






SIZE SOMTOARDIZATION BY 



By C. P. Hirschfeld 



C. H. Berry 



This book is DUE on the last date stamped below 








Southern Branch 
of the 

University of California 

Los Angeles 

Form L-l 

TA 
H G\ 







8061 '12 NW IVd 
'A 'N ' 



junooioqjn 



STHE AMERICAN SOCIETY OF 
MECHANICAL ENGINEERS 

29 WEST THIRTY-NINTH STREET, NEW YORK 



SIZE STANDARDIZATION BY 
PREFERRED NUMBERS 

BY 

C. F. HIRSHFELD 

AND 

C. H. BERRY 



,'THERN 

UNIVERSITY OF CALIFORNtt- 

LIBRARY, 

S ANJ&S.ES. CAklF. 







Presented at the Annual Meeting of The American Society of 
Mechanical Engineers, New York, N. Y., Dec. 4 to 7, 1922 



The Society shall not be responsible for statements or opinions advanced in papers or in 
discussions . . . (B2, Par . 3) 



61315 



SIZE STANDARDIZATION BY 
PREFERRED NUMBERS 

Br C. F. HmsHFELD 1 AXB G. H. Bcuer, 2 DETROIT, MICH. 
Members of the Society 




<** W 

taUes OhaMk At jMtnts Aragb of 

OIZE figures in one way or another in all manufactured 
*-* articles and, in fact, in all articles of commerce. For present 
purposes the word "size" must be interpreted in its broadest 
possible sense. It may indicate any one of the following specifica- 
tions: a purely arbitrary size, such as Model Xo. 1 , Model Xo. 2. 3, 
etc., of a given line of manufactured article; a conventional size upon 
which all manufacturers in a given line have agreed, as sizes of hats 
or shoes; the weight of a package in which a given material is sold; 
, the weight of some arbitrarily chosen quantity, as 10Q4b. rails or 
10-oz. duck; an actual or conventional significant dimension, as 
1-in. round stock or 1-in. lumber; or any one of the numerous di- 
mensions which may be required in the design, fabrication, or mar- 
keting of a given article or manufactured product, 

Viewed from one aspect, size is second only to the product itself 
when dealing with the materialistic side of manufacture and com- 
merce. All manufacture and all commerce are carried on in terms 

Chief, research department. The Detroit Edison Company. Mem. 
A-S.MJ2. 

Engineer, vke-president's office. The Detroit Edison Company. Mom. 
A.SJ4.E, 

Contributed by the Standardization Committee and presented at the 
Annual Meeting, New York, December 4 to 7, 1922, of THE AMERICAN 
SOCIETY OF MECHANICAL ENXUXEER*. All papers are subject to revision. 
3 



4 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

of size. In fact, the need for means of expressing size is probably 
the underlying reason for the heterogeneously assorted systems of 
expression now in use. 

Further, size, in the general sense of the word and also in the spe- 
cific sense, is directly or indirectly made up of two components or 
factors. One of these is a number and the other a dimension, 
as in "1-in." bar stock. 

Numerous cases will be found in which this composite structure 
is not evident, but it will always appear if the search is carried far 
enough. When a given kind of equipment is sold under the size 
designations, Model No. 1, Model No. 2, etc., the model size itself 
is determined by its physical size or capacity. Thus Model No. 1 
may be 4 ft. high, or may have a capacity of 1 ton in a given time, 
while Model No. 2 may be 6 ft. high or have a capacity of 2 tons, 
etc. 

In any general study of sizes we must therefore consider two com- 
ponents which are respectively numerals and units of measurement. 
These units of measurement are matters of custom or use which 
vary both with the type of measurement and with the systems 
adopted in different countries. 

In considering the so-called "Preferred Numbers" we have no 
direct interest in these dimensional units. Our concern is entirely 
with the numerical part of the doublet used for expressing size, 
except in so far as the relations between dimensions in any one 
system of measurement may make certain numerals more or less 
convenient. As an example of the significance of this exception, 
consider the possible means of expressing a length of 27 ft. in the 
English system. This might be called 9 yd., 27 ft., or 324 in. In 
measuring dress goods the yard is the most convenient or at least 
the conventional unit and thus brings about the necessity for the 
use of the numeral 9 in expressing the size indicated. In certain 
other types of measurement, as, for instance, in measuring length 
of lumber, the foot unit is the most convenient and thus makes 
necessary the use of the numeral 27. The inconvenience of using 
the number 324 probably explains the use of feet and yards instead" 
of inches for large measurements. 

Sizes used in industry and commerce may be divided into several 
different categories or classes, as follows: 

1 Sizes which are entirely matters of style, such as the lengths 
of coats, the heights of hats, and other dimensions which change 
from year to year. 

2 Sizes which are determined entirely by personal comfort, such 
as the sizes of men's collars, the sizes of hats and shoes, etc. Each 
of these series of sizes has been worked out by experience and it 
is not unreasonable to assume that at the present time they form 
satisfactory systems. 

3 Sizes which are entirely matters of taste, though not necessarily 
matters of fashion. Thus the proportions of a Doric column enter- 



C. F. HIRSHFELD AND C. H. BERRY 5 

ing into a structure are not determined by strength but by ap- 
pearance. Proportions or sizes of furniture, objects of art, and 
many architectural features fall in this class. 

4 Sizes which are determined by a combination of appearance 
and utility. The sizes of drawer pulls, door knobs and the like fall 
into this class, but for the present they are outside the scope of this 
paper though they may later fall partly or wholly within it. 

5 Sizes which are determined entirely by utility or use value. 
Class 1 sizes are by nature arbitrary and changeable and those of 

classes 2, 3, and 4 are outside the scope of the present paper. Such 
sizes as are grouped under class 5, whether they refer to buckets 
and pails, pots and kettles, bolts, wires, or to any of the innumer- 
able machine parts, fall within the scope of what may be called the 
Theory of Preferred Numbers. 

With the preceding paragraphs by way of introduction, something 
may now be said with respect to this theory, what it is, and what 
it is for. 

THE THEORY OF PREFERRED NUMBERS 

At the present time there is much that is arbitrary in the choice 
of size, even when size is determined by utility or use value, and 
careful study will show that slight variations in the sizes finally 
decided upon would not make a great difference in the use value 
of the pieces. For example, have we any proof that pails should be 
made in 8-, 10-, 12-, and 14-qt. sizes instead of in, say, 9-, 11-, 13- 
and 15-qt. sizes? Or again, when we calculate the required diameter 
of a circular section in a piece of machinery to be 2.237 in., are we 
justified in assuming our calculation so accurate that 2.25 or 2.2 in., 
or possibly even 2 or 2.5 in., will not prove equally suitable? 

Careful study in any drafting room will show that the latitude 
of choice allowed the designer is such that his decision with respect 
to final dimensions is arbitrary within certain limits, and quite often 
within very wide limits. 

In view of these facts it is quite obvious that if certain numerical 
values are universally accepted as preferred values and if they are so 
spaced and" of such extent as to fit in with all requirements met in 
deciding on sizes to be used, the arbitrary choices may be so made as 
to yield sizes expressible in terms of these preferred numbers. More- 
over, if such a thing is possible, very material savings should result 
from its use, some of the most obvious of which are: 

a Mill products which are used in the fabrication of manufactured 
articles could be made in a minimum number of standard sizes so 
chosen as to meet the needs of users who have adopted preferred 
numbers for the sizes of their wares. 

b Measuring instruments and production machinery might be 
simplified and cheapened, because it would be necessary to provide 
for their use with preferred dimensions only instead of providing 
for universal adjustment. 



6 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

c Odd sizes, manufactured through ignorance of real require- 
ments or to meet the supposed, but really illogical, needs of a 
customer or industry, might be eliminated. 

d Life would be made simpler for both the producers and the 
users, because calculation, manufacture, commerce, catalogs, price 
lists, and human memory would deal only with certain easily mem- 
orized and widely used numerals. Certain other advantages will 
become apparent as the subject is further developed. 

One of a practical turn of mind is likely to think that there is 
much of theory in all this and little of practical worth. Sizes have 
been developed by a cut-and-try process and with commercial 
necessity acting as a brake on the overdevelopment of sizes. It 
might be assumed, therefore, that present-day industry is using the 
minimum number of sizes consistent with the meeting of human 
needs and that these sizes are most advantageously chosen. Such 
arguments are weighty and worthy of serious consideration. 

However, systems of preferred numbers have been accepted to a 
certain extent in Germany, and several other European countries 
are indicating an intention of following this lead after having made 
a study of the German systems. It would seem, therefore, that 
we should not complacently accept our present methods and prac- 
tices, but instead give serious thought and study to this problem. 

The authors of this paper are not urging the adoption of a system 
or systems of preferred numbers, but they do urge most emphatic- 
ally a study of the subject in connection with American conditions 
and problems to the end that decision for or against may be made 
with full knowledge of all that is involved. Such a study may 
probably be most conveniently undertaken by considering sizes 
actually in use in this country and the relation which such sizes bear 
to a possible series or to several possible series of preferred numbers. 
Some examples of common cases are given in the paragraphs im- 
mediately following. 

Common wire nails are sold in sizes expressed in terms of "pen- 
nies;" thus we have 2d nails, 3d nails, and so on up to 80d nails. 
These designations originally indicated the price per hundred, but 
now, by arbitrary agreement, they represent certain lengths which 
are expressed in even inches and fractions of inches which have no 
connection whatever with the number used in expressing the com- 
mercial size. The diameters of stock from which the nails are made 
vary from a small value at the low end of the series to a large value 
at the upper end. Presumably the nail is intentionally or unin- 
tentionally proportioned as a long column, and there is therefore 
probably some approximation to a definite relation between diameter 
and length. 

The values of length and diameter for wire nails are plotted 
against the commercial sizes in Fig. 1, from which it can be seen 
that wire-gage sizes and lengths of wire nails are not as well co- 
ordinated as they might be. The nearest convenient commercial 



C. F. HIRSHFELD AND C. H. BERRY 7 

size of wire has been combined with the desired length to give a 
certain size of nail. In fact, half and quarter gage sizes have 
been used in some cases. The effect of this is indicated by the 
jagged line showing strength plotted against size. It is obvious 
that the use of one size of wire for two successive sizes of nails 
results in most erratic variations in strength. The nails are of 
course commercially satisfactory, but this does not mean that 
they are constructed with the minimum use of material or that a 
more satisfactory line of nails might not be developed. This matter 
will be considered later after certain other matters have been dis- 
cussed. 

Wire is sold in certain sizes specified as "gages." Copper wire 
is measured in terms of the American or Brown and Sharpe Gage. 

ITOO 1 




"0 Id 2d 3d Ad 5d 6d Id 8d 9d lOd l?d l6dWdW4Qd 50d60d 80d 
Y/i re-Nail Sizes in Pennies(Nominal Sire) 

FIG. 1 COMMERCIAL SIZES OF WIRE NAILS 

The diameters corresponding to successive gage sizes are plotted to 
semi-logarithmic coordinates in Fig. 2 and it is apparent that this 
series of sizes was developed according to a definite and consistent 
plan. 

Steel wire is measured in terms of the Washburn and Moen, 
Roebling, or Steel Wire Gage. The diameters corresponding to 
successive gage sizes are also plotted in Fig. 2. Evidently there 
is some underlying plan, but it follows what appear to be imperfect 
laws. Apparently a certain law of variation is followed until it 
cannot be followed further. Another is then chosen but later 
abandoned, and so on. There may be good reasons back of this 
peculiar variation, but it seems probable that this gage might have 
been built up upon a much simpler basis. More will be said about 
this in later paragraphs. 



8 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



As another example of sizes in commercial use we may consider 
frying pans. The diameters of one well-known make of frying 
pans are plotted against commercial or nominal sizes in Figs. 3 and 
4. This matter will also be considered later. 




0.0000 



Wire Gage Sijes 
FIG. 2 SEMI-LOGARITHMIC PLOTTING OF WIRE-GAGE SIZES 

Saucepans and preserving kettles are sold in sizes designated in 
terms of quarts. Saucepans are made in comparatively small 
sizes and preserving kettles in comparatively large sizes, but the 



C. F. HIRSHFELD AND C. H. BERRY 



s 





-jo 

- 9 
I 

a 



0001 2345678 
jmbers (Nominal Sire) 

FIG. 3 



eter, 

-1^ in 



00 12 34 5 6 75 
Size Numbers (Nominal Size) 
FIG. 4 

PIGS. 3 AND 4 DIAMETERS OF FRYING PANS PLOTTED AGAINST COMMERCIAL 
OR NOMINAL SIZES 



10 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



K 
II 
10 
9 
8 



I 6 
1 s 

3 
2 

1 

, 






































/ 


















I 






































1 




















/ 


















/ 


















/ 


















/ 


















/ 
















X 


^ 






































\.&CDEFGH 1 J K 



Si^eHumber (NominaiSije) 
FIG. 5 



Capacity , Quarts 
ni w -N tn <T>-jcoy>Sr3 


















X 


















-?*- 


















^ 




















fT 


















s 


















s 


















X 


















/ 
















/ 


/ 



















ABCDEF6JIJJK 

Sije Numbers (Nommal 6136) 
FIG. 6 

FIGS. 5 AND 6 CAPACITIES OF SAUCEPANS PLOTTED AGAINST COMMERCIAL 
OR NOMINAL SIZES 



C. F. HIRSHFELD AND C. H. BERRY 



11 



two overlap so that sizes of equal capacity can be obtained in the 
larger sizes of saucepans and in the smaller sizes of preserving kettles. 
The two kinds of utensils are interchangeable in use, so that it is 
probably not illogical to consider them together. The sizes avail- 
able in one make are plotted to ordinary coordinates in Fig. 5. 



20 40 
30 

10 20 

5 10 
4 I 

3 6 

5 

5 5 
.*T 2 

0.4 o o.l 

0.7 

0.3 0.6 






1 








\ l 


5.0 
4.0 
3.0 

2.0 
1.0 
0.500 
0.250 
0.125 

0.0125 


















/ 



















/ 
















I 


/ 
























































^ 




















j 




















/ 




















J 


















y 






































z 


























































/ 




















/ 




















/ 


















0.2 0.4 
0.3 

0.1 0.2 
0.05 0.1 

J 


} 




















/ 




















\f 








































vBCDEFGHIJK 



Size Number (Nominal Si^es) 

FIG. 7 CAPACITIES OF LIQUID MEASURES PLOTTED AGAINST NOMINAL 
SIZES 

The same values are plotted to semi-logarithmic coordinates in 

Fig. 6. 

Liquid measures are available in numerous sizes, all commonly 
designated in terms of capacity. The sizes obtainable from one 
well-known maker are plotted to semi-logarithmic coordinates in 
Fig. 7. 

Commercial sizes of steel shafting are shown in Fig. 8. 

Bolts have been standardized in different ways and for different 



12 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



purposes by various organizations. One very complete standard 
is that of the Society of Automotive Engineers. The diameter, 
area of stock, and area at root of thread for S.A.E. Standard bolts 
are plotted to semi-logarithmic coordinates in Fig. 9. A similar 
set of graphs for the United States Standard is given in Fig. 10. 

Pipes are well standardized in several different weights. They 
are sold in terms of nominal diameters and the actual diameters 
practically never equal the nominal diameters. The actual inside 
diameters and areas of standard full-weight wrought-iron pipes 




Successive Commercial Sijes 

FIG. 8 DIAMETERS AND AREAS OF STEEL SHAFTING PLOTTED AGAINST 
SUCCESSIVE COMMERCIAL SIZES 

from Vg in. to 10 in. nominal diameter are plotted to semi-logarith- 
mic coordinates in Fig. 11. 

A brief survey of Figs. 2, 4, 6, 7, 8, 9, 10 and 11, all of which show 
sizes plotted to semi-logarithmic coordinates, will indicate a sur- 
prisingly large number of cases in which successive sizes fall very 
nearly on one straight line or on two or three straight lines of slightly 
different slopes. If space permitted many more examples of the 
same thing could be given. 

This peculiar tendency naturally suggests some underlying law 
of size or size variation. If sizes tend to fall on such straight lines, 
it follows that the mathematical characteristics of such lines must 
express what we may call the natural law of size variation. 



C. F. HIRSHFELD AND C. H. BERRY 13 

THE APPLICATION OF THE THEORY OF GEOMETRICAL SERIES 

On semi-logarithmic paper a straight line which is not parallel 
to either axis is the plot of a geometrical series. That is, such a 
line represents a succession of terms each of which bears a constant 
ratio to the preceding one. The following succession of terms 
represents part of a geometric series: 



a, 2a, 4a, 8a, 16a, 



etc. 



2.0 




0.01 



Successive Si^es 
FIG. 9 S.A.E. STANDARD BOLTS 
Such a series can be written symbolically, as: 

a, ra, r 2 a, r 3 a, r n ~ l a 

in which 

a = first term of the series or the number on which it is built up 
r = ratio of each term to the preceding term, and 
n = the number of the term in the series. 
German scientists and technicians have investigated this matter 
of size variation, including several geometrical series which may 
be used as expressions of the law of size variation. While they 
have not yet formally adopted them, they are leaning very markedly 



14 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



toward a system of series with ratios equal to the 5th, 10th, 20th, 
40th, and 80th roots of 10 and have, in fact, adopted it tentatively. 




Successive Sijes 
FIG. 10 U. S. STANDARD BOLTS 

Such series would be simply expressed as follows for the lOth-root 
series: 

a, \/io a, (>/To) 2 a, (\/Ib> a, etc. 



C. F. HIRSHFELD AND C. H. BERRY 



15 



which is equal to 

a, 1.259a, 1.58a, 1.99o, etc. 
For the 20th-root series: 

, (\/10) 2 a, (\/10) 3 a, etc. 



a, 



3 




I 



n "..? 

0.4 

<os 

0.2 



0.1 



Successive Sijes 

FIG. 11 STANDARD WROUGHT-!RON PIPE 
which is equal to 

a, 1.122a, 1.259a, I Ala, etc. 

The whole idea can be illustrated most clearly by dealing with 
the series which has a ratio equal to a \/10, i.e., 10 l/1M . Such 
a series built upon the number 1, that is, with a equal to 1, is drawn 
to semi-logarithmic coordinates in Fig. 12. 



16 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



Assume that it is desired to make a given article in eleven sizes 
progressing as in this series. The size designations are indicated 
as A, B, C, D, etc., in Fig. 12. The dimensional sizes are given 
by the values of the ordinates above the size designations. If a 
smaller number of models equally distributed over the same range 
of dimensional sizes is required, we might use sizes A, C, E, G, I, 
and K. But such a choice would be exactly equivalent to using 
a series such as that shown at the left in Fig. 13, which is a series 
with the ratio \/W. Or if we desired only three sizes say, A, F, 



CJ 

in 













































s 
























/ 




j 


















s 
























\s 






<0 c 












j 




/ 








'? 1 










' 


<\ 


/ 










1- 










g 


/ 












E 3 

"o 








/ 


/ 














lc 


/ 


r 


/ 



















VCDEFGHIJKL 

Successive 61366 

FIG. 12 GRAPH OF SIZE SERIES WITH RATIO 

and K, this would be equivalent to using the \/10, or a series 
such as that shown to the right of Fig. 13. 

Bearing these ideas in mind, inspection of Fig. 14 will show that 
a series with ratio \/10 makes available only two of the sizes here 
under consideration, namely, the first and the last. A series with 
ratio \/10 gives three sizes, the same two extremes and one inter- 
mediate corresponding to F of the original arrangement. A series 
with ratio \/10 yields four sizes, the two intermediates not cor- 
responding to previous sizes because 30 is not evenly divisible into 
100. A proportionate number of sizes is obtainable with the series 
with ratios equal to \/10, \/io, \/10, \XlO, and \/lQ. The 

series with ratio equal to \/10 yields six of the original sizes as 
previously indicated. 



C. F. HIRSHFELD AND C. H. BERRY 



17 



Going back now to the original series with ratio equal to 
shown in Fig. 12, let us assume that the 11 sizes designated by letters 
A to K, inclusive, are not sufficient, that is, the finer subdivisions 
are required. These can be obtained by inserting sizes midway 
between A and B, B and C, C and D, etc., as shown in Fig. 15. 
We should then get sizes which might be designated as A, A 1 /^ B, 
B 1 / 2 , C, C l / 2 , etc., and the dimensions of each size would be related 
to the dimension of the preceding size by the same ratio, but this 
ratio would not be the same as that in the case first considered. 

The last statement is likely to be a bit puzzling to those who do 



10 
9 
8 
7 

05 & 

5 

4 

O) 

|3 
2 

1 


































/ 






















/ 












/ 












7 












/ 










. 














/ 








. 


f n / 














/ 








SZ 














/ 








m 














jil 


/ 






A 


/ 














1 






/ 


/ 
















7 







A C E 6 I K 

Successive 



A F K 



FIG. 13 GRAPHS OF SIZE SERIES HAVING RATIOS 5(l \/10 AND 2 -\/10 

not handle such series frequently, but it is easily explained. The 
line as drawn yields a value of 10 as ordinate for 100 of the smallest 
divisions on the horizontal axis. But 10 is the 100th power of 
1 \/lQ. The ordinate of the first small division counting from the 
left on the horizontal axis is therefore equal to the ordinate at A 
multiplied by 1 \/10; the ordinate of the second small horizontal 
division is the ordinate at A multiplied by 1 2/io x '\/10, or 
^X/IO) 2 ; the ordinate at B is equal to the ordinate at A multiplied 
by ('x/lO) 10 ; the ordinate at C is equal to the ordinate at A multi- 



plied by C\/lOy o ; and so on. It is therefore obvious that 
Ordinate at B = Ordinate at A X C\/10) 10 
Ordinate at C = Ordinate at B X C\/10) 10 , etc. 



18 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



but 

Ordinate at A 1 /* = Ordinate at A X (^lO) 5 , and 
Ordinate at B = Ordinate at A l / z X ('x/lO) 5 . 
That is, the ratio between successive sizes of the A, B, C, D, etc., 
series is (*\/10) 1() = \/10, and the ratio between successive 
sizes of the A l / 9 , A, B, B l / z , etc., series is C\/1Q) 5 =\/W. If 
desired, the subdivision can be carried to any extent, but no matter 
how far it is carried the same characteristic relations will hold true. 




C D E F G H J 

Successive Si^es 



J K L 



FIQ. 14 SIZES AVAILABLE WHEN SERIES WITH DIFFERENT RATIOS ARE 
EMPLOYED 

If the requirements to be met are such that comparatively small 
variations of size are required in the smaller sizes and larger varia- 
tions in the larger sizes, we might choose an arrangement such as 
that shown in Fig. 16. This appears to be a very minor variation 
of what has preceded, but in fact it is quite a major variation. The 
series of sizes resulting from such uneven subdivision no longer has 
the same, characteristics as were described in connection with the 
sizes A, B, C, D, etc. The ratio between successive sizes changes 
at E, H, I, and /. This is shown clearly by plotting dimensions 
from Fig. 16 with nominal sizes evenly spaced as in Fig. 17. The 
ratios are indicated on each part of the broken line of this figure. 

From what has been done in developing the graph in Fig. 16, 
it is apparent that its shape will vary with the way in which one 
chooses to distribute nominal sizes among the possible evenly dis- 



C K. HIRSHFELD AND C. H. BERRY 



Dimensions 
s^~ ^ CM 4* m o-jootoo 
















































































r~ 






































_s 




































s 







































*/ 




































s\ 




































s 


/ 
































. 


f 


s 






























/ 


/ 


/ 


























' 


/ 


/ 


' 
































[AiBBjCCiD DiEE?F F? & G?H H?l',J? J Jj>KK t 

Successive Sjes 



FIG. 15 SHOWING METHOD OF INTERPOLATING SIZES IN A SERIES 



Dimensions 
v^ ro OJ -t CTI <s> -joacDo c 


















( . 




















^s 




















s^ 




















s^ 




















S^ 




















y^ 


















^/ 


















. 


S 
















' 


/ 








d 


< 


/ 


' 


... 








, , 




[A?BB?C V E F H K 



Successve 

. -:-: ;:: " 
t^ic. 16 nUiTEVEN SUBDIVISION OF SIZES 



Stte 

'' 



20 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



tributed sizes. If the nominal sizes be closely spaced with refer- 
ence to possible sizes with any assumed even increments, the re- 
sultant line will be less steep than if the nominal sizes are more 
widely spaced among the possible sizes. 

Comparison of Figs. 12 to 17, inclusive, with Figs. 2, 4, and 6 to 
11, inclusive, will show sufficient resemblance between the two 
groups to lead one to suspect that possibly the geometrical series is 
an expression of the law which underlies variation of size in articles 
whose size is determined by use value. One may object to such a 
conclusion by pointing out that in many cases in which actual sizes 



20 



o 5 

'1 4 
a) 




i 



A A.? B 'gr C I> E F ff K L 

Sycc^ssive 61355 

FIG. 17 DIMENSIONS FROM FIG. 16 PLOTTED WITH NOMINAL SIZES EVENLY 
SPACED 

are plotted they do not fall exactly on the straight lines which have 
been drawn, and, further, that the use of a logarithmic scale distorts 
the significance of vertical departure from the lines so that it is 
possible that a small departure on the plot may mean a large dis- 
crepancy in actual size. 

PRESENT SYSTEM SIMILAR TO PREFERRED NUMBERS 

These criticisms are valid but they do not vitiate the suggested 
conclusion. Rather peculiarly, this is true because we are already 
more or less unconsciously using a set of preferred numbers. We 
use certain fractions of inches for dimensions smaller than one inch 
in preference to all other possible fractions. Thus our first choice 



C. F. HIRSHFELD AND C. H. BERRY 



21 



is 1 / 2 in. ; the next, 1 / t and 3 /4 in. ; the next Vs, 3 A> 5 A, and so on. 
For dimensions larger than one inch we do the same sort of thing 
but in two different ways. Our choice with respect to fractions 
of one inch is the same as before, namely, a system based on eight 
with preference given to 4 / 8 ; 2 A, <A, and 6 A; y g , /, 3 A, 4 A, 5 A, 6 A, 
and 7 A; etc., in the order indicated by groups. But for fractions 
of one foot we use a system based on twelve and, whether we express 
ourselves in inches or in fractions of a foot, we give preference to 
Vis; Vi2, Vis, Vis; Vis, Vis, Vis, Vis, 10 A 2 ; etc., in the order indicated 
by groups. 

Therefore, in actually setting the dimensions which shall char- 
acterize different sizes of manufactured products, we generally 



.inn 










- 


__^ 


^ 


^ 




Diame+erof Pan,Sq. In 
_o S'S_gSS38J 






^^ 


^5=S5= 


, * 




























/ 


















/ 











































































I 5 4 5 6 7 8 



Nominal Sije of Pans ' 

FIG. 18 SQUARES OF DIAMETERS OF FRYING PANS PLOTTED AGAINST 
NOMINAL SIZES 

choose on the basis of these preferred numbers, using that one which 
happens to fall nearest to our desire in each case. 

Consider, for example, the dimensions of frying pans as plotted 
in Fig. 3. Diameter is certainly the most significant dimension of 
a frying pan. After that would come depth, and after that the 
combination of dimensions which affect ability to stack different 
sizes or a number of the same size. The smallest diameter is 6 in., 
a very convenient number, actually the preferred 6 /i2 or 1 / 2 . This 
size is used for such purposes as frying a single egg, a single chop, 
or other small quantities of food. The next size has a diameter of 
8 in., another one of our unconsciously preferred numbers, and 
beyond that diameters increase by half-inches to a diameter of 10 
in. Beyond this size the diameters increase by 1-in. increments. 

If there is any significance to the diameter of a frying pan it must 
result from the fact that the area is proportional to the square of 
that dimension. The squares of the diameters, and therefore 
figures proportional to cooking area, are plotted against nominal 
sizes in Fig. 18. The most casual inspection shows that in all 
probability no law underlies the choice of these diameters, and one 



22 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

is driven to the conclusion that sizes of about the proportions made 
have been found useful and that they have been chosen to fall on 
the preferred half-inch and whole-inch intervals. 

Let us assume for the sake of argument that a 6-in., an 8-in., 
and a 13-in. size are necessary and that the cooking area is the 
underlying, governing criterion in choice of size. Let us also as- 
sume that six sizes are required between the 8-in. and the 13-in. 
sizes. The simplest possible arrangement would result if we used 
diameters consistent with a straight line drawn through the 8-in. 
and 13-in. points in Fig. 18. The resultant areas and diameters 
are given in Table 1 for ready comparison with the actual com- 
mercial diameters. 

TABLE 1 COMPARISON OF CALCULATED AND COMMERCIAL 

DIAMETERS OF FRYING PANS 
Nominal Size Area Diameter Commercial Diameter 

64.0 8.00 8.0 

1 73.5 8.56 8.5 

2 84.0 9.16 9.0 

3 96.5 9.83 9.5 

4 111.0 10.53 10.0 

5 128.0 11.31 11.0 
. 6 147.0 12.12 12.0 

7 169.0 13.00 13.0 

It is at once apparent that the diameters corresponding to the 
smooth line of Fig. 18 are most inconvenient numerical values in 
comparison with the more commonly used numerical values which 
represent the actual commercial diameters. On the other hand, 
it is probably true that the resulting sizes are far more rational and 
would prove of greater use value, if there were any means of mea- 
suring such use value, since each size bears a certain definite constant 
relation to that which precedes it and that which follows it in the 
series. It is possible that a still better result could be obtained by 
using two different slopes between the two extremes so that di- 
ameters increased more slowly at the start and more rapidly later. 
Such an arrangement would correspond more nearly to present 
commercial sizes. 

Now suppose for a moment that instead of using such values as 
even inches and half-inches we had at some time arbitrarily decided 
to give preference to, let us say, the curious numbers 8, 8.5, 9.2, 
9.8, 10.5, 11.3, 12.1 and 13. If, under those circumstances, we 
developed a set of diameters for frying pans as given in the third 
column of Table 1 we should have driven ourselves into the situa- 
tion illustrated by Table 2. 

TABLE 2 COMPARISON OF CALCULATED AND PREFERRED 

DIAMETERS OF FRYING PANS 

Calculated Diameter Preferred Numbers 

.8.00 8.0 

8.56 8.5 

9.16 9.2 

9.83 9.8 

10.53 10.5 

11.31 11.3 

12.12 12.1 

13.00 13.0 



C. F. HIRSHFELD AND C. H. BERRY 23 

Obviously, we would not hesitate to round out the calculated 
values to the preferred values, and the frying pans built upon the 
preferred-number diameters would probably have just as great a 
use value as would pans built upon the calculated diameters. 

As another example of the use of preferred numbers, consider 
the matter of wrought-iron and steel pipe sizes. This material is 
sold in several different "weights" or wall thicknesses, additional 
thickness being obtained in most cases at the expense of internal 
diameter. In order to simplify matters to the greatest possible 
extent only one weight will be considered, namely, Standard Full 
Weight, and only such sizes will be used as are commonly sold on 
the basis of nominal inside diameter, that is, sizes up to and includ- 
ing 12-in. pipe. The sizes in which such pipe is graded are desig- 
nated in inches and fractions of inches and refer to a nominal inside 
diameter, not to a real diameter. Table 3 will indicate the extent 
to which the nominal diameter is purely a fictitious designation. 

TABLE 3 NOMINAL AND APPROXIMATE ACTUAL PIPE SIZES 

Nominal Diameter, Decimal Equivalent of Approximate Internal 

in. Nominal Diameter, in. Diameter, in. 

i/s 0.125 0.269 

1/4 0.250 0.364 

Vs 0.375 0.493 

i/j 0.500 0.622 - ' 

V4 0.750 . 0.824 

1 1.000 1.049 
li/4 1.250 1.380 
li/2 1.500 1.610 

2 2.000 2.067 
21/2 2.500 2.469 

3 3.000 3.068 
31/2 3.500 3.548 

4 4.000 4.026 



10 10.000 10.192 

11 11.000 11.000 

12 12.000 12.090 

It is of interest to note that while we purchase such pipe in terms 
of even inches and fractions of inches, what we really purchase is 
pipe having internal diameters designated by figures which are 
apparently just as inconvenient as any of those which were worked 
out for frying-pan diameters. We are thus in fact using figures of 
this same form and sort but disguising them in order to express 
sizes in our preferred-number system. 

Pipes are used principally as conveyors of liquids and gases and 
the significant dimension is the internal diameter, since this deter- 
mines the cross-sectional area and therefore the carrying capacity 
of the pipe. Inspection of Fig. 11 shows a rather ordered progress 
of cross-sectional area from the smallest to the largest pipe sizes. 
It is probable that there was originally some reason back of the 
variations from exact order in successive sizes and areas. Possibly 
it had something to do with thickness of material available, or with 
some necessary relation between internal and external diameters, 



24 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



or, more remotely, it may have had something to do with strength 
against bursting. 

However, it should certainly be possible to make pipe to areas 
such as those indicated by the straight lines drawn on Figure 19. 
If this were done, the actual internal diameters would be as indicated 



ioo 










































J 






00 


70 
60 




































-f 


^ 










DO 


40 


































/ 
















Tn 
































/ 
















30 


?fl 






























/ 




















10 
























\ 


^ 










| 


6 


^ 


^ 






10 


6 
5 
4 


















t 


/ 




it; 


^ 


P H 


^ 


















tj 
5 
4 


7 














i 


y 1 


/ 




s> 


/ 
























3 


l] 












1 




/ 


/ 


/ 






























" 














t- 


































^ o 

'0 fe 


i/Se 

s"85 






/ 


/ 








































0.6 n 

0.5 5 
04 c5 


< 031 


^ 






/ 








































nz 


no 






7 










































0? 


0.1 




/j 












































O.I 


0.06 
0.05 
004 






















































































































































nni 



















































Succe&eive Cor 



jl Sijes 



FIG. 19 STANDARD WROUGHT-!RON PIPE 
(Points represent actual diameters and areas as now made.) 

by the other series of lines on the same figure. Inspectionjof the 
locations of the points representing the actual present-day diameters 
with reference to their proximity to the new-diameter lines will 
show that the new diameters required will not vary greatly from 
those actually in use. The variation appears to be of a much smaller 
order than was found in the study of frying pans. 

If one objected to designating pipes in terms of these odd actual 



C. F. HIRSHFELD AND C. H. BERRY 25 

internal diameters, it would be perfectly possible to use the same sort 
of fiction which we now use under exactly the same conditions. 

Examples of these sorts could be produced in almost endless 
variety, but it seems as though those just cited are sufficient to 
illustrate the points under discussion. We do use preferred num- 
bers now, and very often these preferred numbers are used in a 
purely nominal sense while the real size dimensions are expressed 
in unwieldly decimal fractions. In some cases this is so extreme 
that we give these inconvenient decimal fractions special designa- 
tions so that we will not have to deal with the numbers themselves. 
Such a case has just been considered in connection with pipes. 
Many others exist. Thus a No. 22 copper wire means something 



.Mil I I I I 

2 in. -m Klrr.-^-2/n. 
7/7.- > 



/#.- 



FIG. 20 LENGTH OF ONE FOOT so SUBDIVIDED THAT LENGTHS OF DIVISION 
LINES REPRESENT ORDER OF PREFERENCE 

to almost every engineer, and yet there are very few who know that 
a No. 22 copper wire has a diameter of 0.0253 in. And why should 
they when the convenient round number 22 meets all ordinary needs 
and when exact diameter and area can always be obtained from a 
gage table if needed? 

Unfortunately our preferred numbers are most irrationally 
related from many points of view. This follows directly from our 
conventional use of the inch and foot and from the way in which 
our system of numbers is built up. The effect of the inch and foot 
is illustrated in Fig. 20, in which a length of 12 in. is laid out to an 
arbitrary scale with certain preferred subdivisions set in in such a 
way that the length of the subdivision line represents the order 
of preference. It is apparent that any series of sizes which starts 
with dimensions less than one inch and ends with dimensions ex- 
pressed in feet, inches, and fractions of an inch must almost certainly 
be made up of steps of most varying mathematical character. 

CONSTRUCTION OF GEOMETRICAL SERIES 

Again, our system of numbers is itself basically peculiar. The 
logical steps of progression from 1 to 10 are by units or by simple 
or fractional multiples of units. The tendency is always toward 
even steps and there are surprisingly few possibilities. Thus we 
may use the following: 

1, 10; 1,5, 10; (1,2.5, 5,7.5, 10); 

1, 2, 4, 6, 8, 10; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 



26 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

These series are all arithmetical, and any attempt to arrange a 
geometrical series between 1 and 10 leads immediately to com- 
plicated, or at least inconvenient, decimal fractions. 

Above 10, conditions are somewhat better in that there are nu- 
merous geometrical series yielding simple numbers, such as 
10, 20, 40, 80, 160, 320, etc. 
10, 30, 90, 270, 810, etc. 
10, 40, 160, 640, etc. 
10, 50, 250, 1250, etc. 

but it is obvious that the increase of value in the first series, which 
is characterized by the slowest rate of increase, is exceedingly rapid. 

With the exception of such geometrical series one is limited to 
arithmetical series in developing steps of such character as to yield 
simple even numerals. One may progress by ones, twos, threes, 
fours, fives, etc., up to any desired value and thus develop any num- 
ber of jarithmetical series. But any attempt to develop geometrical 
series other than those just indicated leads immediately to fractional 
numbers. 

These are the facts with which we are confronted, and, even if it 
be assumed that we can and possibly may change our system of 
measurement, it seems foolish to suppose that we will ever change 
our system of numbers. It looks, therefore, as though we may as 
well make up our minds to do the best we can with what we have. 

The Germans have an official system of measurement which 
coincides in arrangement with the system of numbers. They there- 
fore seem to have only one problem to solve where we have two. 
The truth of the matter is that in many cases they also have two, 
but there are enough cases in which they have only the one so that 
they can confine their attention largely to those cases for the time 
being. The movement to adopt preferred numbers may be regarded 
as an effort to evolve a number system better suited to our tech- 
nical needs than the present system. Of necessity it is expressed 
in terms of our present system and as a result it appears at a dis- 
advantage. 

The Germans have two sets of preferred numbers and, the authors 
are informed, these numbers are now in use to a limited extent. 
The first is a set of "standard diameters," and was adopted before 
the recent development of the more general "preferred numbers." 
The second is built up on the 80th root of 10, and is like the one dis- 
cussed in connection with Figs. 12 to 17. The method of con- 
struction is best illustrated by writing down the numerical terms 
of such series. These values are given in Table 4 for all series 
between that with ratio \/I6 and that with ratio \/10. 

A casual inspection of Table 4 shows that the series with ratio 
\/To contains all the terms comprised in the series with ratio \/10 
and in addition one term which is the geometrical mean between 
every two of those in the simpler series. Similarly, the series 



TABLE 4 PREFERRED-NUMBER SYSTEM ADOPTED IN GER 
MANY EXACT VALUES 



Ratio = Ratio = Ratio = 


Ratio 


_ 




Ratio = 




S/10 = 1.585 10 \/10 = 1.259 2 VlO = 1.122 


<VIo 


= 


1.059 


80 VIo = 


1.029 


Numer- Numer- Numer- 
Term ical Term ical Term ical 
Value Value Value 


Term 


Numer- 
Value 


Numer- 
Term ical 

17<t1<io 


1.000 1.000 1.000 





1 


000 





.000 










J 


029 




i 


1 


059 


2 


.059 


.'.'.','. 1 1.122 


2 


1 


i22 


3 

4 


.090 
.122 












155 




's 


1 


isg 


6 


.189 










7 


.123 


'.'. '.'.'.'. 'i l!259 '2 l!259 


'4 


1 


259 


8 


.259 










g 


296 




5 


1 


334 


10 


.334 










11 


373 


.... 3 1.413 


6 


1 


4is 


12 


.413 










13 


.454 




"7 


1 


496 


14 


.496 










15 


.540 


1 l'.585 '2 l'.585 '4 l'.585 


's 


1 


585 


16 












17 


.631 




'g 


1 


679 


18 


.679 










19 


728 


'.'. '.'.'.'. '.'. '.'.'+ '5 1 ! 778 


10 


I 


778 


20 


'.778 










21 


830 




ii 


1 


884 


22 


884 










23 


939 


.... 3 1 . 995 6 1 . 995 


12 


1 


995 


24 






is 


2 


iis 


25 2.054 
26 . 2.113 










27 


J 175 


!! '.'.'.'. '.'. '.'.'.'. '7 2'.239 


14 


2 


239 


28 


2.239 










29 


2.304 




is 


2 


371 


30 


2.371 










31 


2 441 


2 2.512 4 2.512 8 2.512 


ie 


2 


5i2. 


32 


1.512' 










33 


2.585 




17 


2 


eei 


34 


2.661 










35 


2 738 


'.'. '.'.'.'. '.'. '.'.'.'. 9 2.'818 


is 


2 


818 


36 . 


2^818 










37 


2.901 




19 


2 


985 


38 


2.985 


'.. .... 5 3.162 i6 3.ie>2 


20 


3 


i62 


39 3.073 
40 3.162 










41 


5 255 




2i 


3 


350 


42 


5.350 










43 


5 447 


'.'. '.'.'.'. '.'. '.'.'.'. ii 3'.548 


22 


3 


548 


44 


I.U48 










45 ; 


i 652 




23 


3 


758 


46 ; 


J.758 










47 J 


.868 


'3 s.'gsi 'e 3'.98i 12 3'.98i 


24 


3 


98i 


48 ; 


.981 










49 * 


.097 




25 


4 


217 


50 4 


.217 










51 4 


340 


'.'. '.'.'.'. '.'. '.'.'.'. 13 4'.467 


26 


4 


467 


52 4 


!467 










53 4 


.597 




27 


4 


732 


54 4 


.732 










55 4 


.870 


'.'. '.'.'.'. ~7 2'.015 14 5.Q12 


28 


5 


012 


5& i 


.012 










57 i 


.158 




29 


5 


309 


58 i 


.309 










59 i 


.464 


'.'. '.'.'.'. '.'. '.'.'.'. io fl!623 


30 


5 


623 


60 i 


.623 










61 i 


788 




si 


5 


957 


62 


.957 










63 e 


131 


'4 e.'sio 's s'.sio ie G'.SIO 


32 


6 


sio 


64 


.310 










65 6 


494 




33 


6 


683 


66 6 


.683 










67 6 


879 


'.'. '.'.'.'. '.'. .... i? 7^679 


34 


7. 


079 


68 7 


.079 










69 7 


.286 




35 


7. 


igg 


70 7 


.49g 










71 7 


718 


;; ;;;; '9 7^943 is 7.943 


36 


7. 


g43 


72 7 


.g43 










73 8 


.175 




37 


s'. 


iii 


74 8 


.414 










75 8 


660 


;;" . ;;;;;; '.'.'.'. ig s^is 


38 


8.' 


913 


76 8 


!913 










77 9 


.173 




39 


9 . 


441 


78 S 


.441 










79 fl 


71fi 


5 10 ! 666 io 10 ! 666 26 10.666 


40 


10! 


666 


80 10.000 



28 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

with ratio \/10 contains all those in the series with ratio \/10 
with one additional term between every two of those occurring in 
the latter series. The same sort of thing must hold, no matter how 
far one carries the construction of such series. 

Obviously, if one wished to construct a set of models with a small 
number of steps or sizes, he would use some or all of thejvalues in 
the \/10 series, or possibly even drop back to the \/10 series. 
If more sizes were wanted the Z ^/\Q ser ies or the \/W or the \/H) 
would be used as required. 

This gives several sets of preferred numbers, all of which, however, 
belong to one family. 

The Germans then take one further step and round off these 
preferred numbers, making 1.259 into 1.2; 1.585 into 1.6; etc. This 
gives the final set or sets of preferred numbers. Such sets are 
illustrated in Table 5, from which the method of construction will 
be obvious when studied in connection with Table 4. 

TABLE 5 PREFERRED-NUMBER SYSTEM ADOPTED IN GERMANY- 
SIMPLIFIED VALUEvS 

-Values from 50 to 500 

vSeries 1 Series 2 Series 3 Series 4 
50 50 50 




190 
200 
210 



The authors believe that it may be a mistake to round out the 
numbers in these tables instead of preserving the original values. 
The original values are the exact values, and the extent of rounding 



C. F. HIRSHFELD AND C. H. BERRY 29 

in some cases is so great as to entirely mask the original value. If 
one regards the adoption of preferred numbers as the adoption of 
a new number system, much can be said in favor of preserving the 
peculiar decimal fractions. However, only experience can prove 
the correctness or incorrectness of such practice. 

APPLICATION OF PREFERRED-NUMBERS SYSTEM TO UNITS OF 
MEASUREMENT 

Any attempt to apply such a system of numbers to our units of 
measurement immediately introduces a complication. The decimal 
fractions do not lend themselves readily to use with feet and inches 
in the way in which we now use, or think we use, those dimensions. 
However, it has been shown already that in many cases we are now 
dealing with decimal fractions in some of our most common articles 
of commerce, and there is no reason to suppose that we could not ex- 
tend this practice if the results to be achieved warranted it. If we 
adopted the inch as a standard of length, for instance, and used 
decimal fractions of inches and multiples by tens we should have a 
system with many of the conveniences of the metric system. 

After all, commerce is actually conducted as much in terms of 
nominal sizes as in terms of actual or approximate dimensions, so 
that no difficulty need be anticipated in that direction. Production 
at the present time is effected largely in terms of gages, and it is 
certainly just as easy to produce gages to check a dimension equal to, 
say, 1.585 as it is to construct a gage to check a dimension equal to 
1 .750, that is, ! 3 / 4 in. or 1 ft. 9 in. It would seem as though no great 
difficulties would be introduced into production or manufacture 
by the adoption of such decimal fractions. The only other function 
needing consideration is that of design. The designer in most cases 
works in terms of decimal fractions anyway, and with our present 
system is confronted with the necessity of converting his final re- 
sults into the conventional fractions of an inch. Certainly he 
ought not to complain if a system of measurement expressed in 
decimal fractions is adopted. 

Let us now return to the graphs of actual sizes of commercial 
products as given in Figs. 1 to 11, 18, and 19. Study of such 
graphs will show that those products which are largely used as 
components of manufactured or fabricated articles, such as bar 
steel, bolts, structural shapes, wire, etc., are generally made in 
such sizes that the graph to semi-logarithmic coordinates is con- 
cave toward the horizontal axis. This means that in the larger 
sizes such materials are made to finer sizes (smaller steps) in a 
geometrical sense than they are in the smaller sizes. On the other 
hand, in the case of finished articles such as household utensils, 
containers, machines of various sorts, etc., there is a greater ten- 
dency for the graph to flatten out or to become convex toward the 
horizontal axis. This indicates a tendency toward more uniform 



30 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

spacing throughout the line and in some cases toward wider spacing, 
in a geometrical sense, in the larger sizes. 

Such tendencies as these may not be lost sight of in any effort 
to rationalize sizes by the adoption of preferred numbers. It may 
be that we now produce an excessive number of sizes in certain 
products and that we would actually save in the long run by pro- 
ducing fewer, but such matters require long and detailed study 
before conclusions can be drawn. 

There is a further thought which should be kept in mind during 
all efforts to adopt preferred numbers. Materials as used in pro- 
duction are basically dimensioned to meet certain loading re- 
quirements. It happens that physical dimensions appear in several 
different ways in the formulas which are involved, particularly 
with reference to their exponents. 

As an illustration of the significance of this, strength in tension 
or compression varies with the square of the diameter in the case 
of a solid circular section, but strength with respect to bending 
varies as the cube of the diameter. If we imagine a set of preferred 
numbers in use and further imagine that round steel stock is rolled 
to preferred diameters, it hardly seems possible that one set of pre- 
ferred diameters can give equally desirable variations of its squares 
and its cubes. When one considers the further complications which 
are introduced when other types of loading are brought into consid- 
eration, the case appears complex indeed. 

Such difficulties may prove to be more apparent than real, but the 
underlying ideas should be borne in mind in any detailed study of 
this subject. 

POSSIBILITIES IN SIMPLIFICATION OF DESIGN 

Attention thus far has been concentrated largely on size as a 
finished or completed proposition with little reference to the mechan- 
ism of design. No small part of the saving to be expected should 
accrue from simplification of design. This matter has been spar- 
ingly treated in the rather meager literature of this subject and a 
few very interesting studies have been recorded. In particular, 
a paper by Erich Hoffman published in Mitteilungen des Normen- 
ausschuss der Deutschen Industrie, February, 1920, contains two very 
interesting examples, one of these being a set of eyebolts. 

The collar diameter of the smallest eyebolt is taken as the basic 
dimension, and for successively larger bolts this is increased in 
the ratio \/10 for the smaller range of sizes, and in the ratio 2 \/To 
for the larger range of sizes. Other dimensions of each eyebolt are 
obtained by taking its collar diameter as a starting point and multi- 
plying by factors differing with the dimension sought. The fac- 
tor used for a given dimension (such as internal diameter of eye), 
however, is the same for all sizes of bolt, and the result is therefore 
a set of geometrically similar eyebolts. As a matter of fact, the 
Germans have gone one step further, in that, for the set of eye- 



C. F. HIRSHFELD AND C. H. BERRY 31 

bolts in question, each dimensional multiplying factor is taken as 
one term in the series with the ratio \/10, and thus each of these 
factors is itself a preferred number. 

This sounds like a very pretty piece of mathematical juggling in 
connection with machine design, but it really has a very deep practi- 
cal significance. A designer need design and draw only one size 
of eyebolt if he is sure that a certain series can be applied. All 
other desired sizes can then be obtained by the simplest form of 
calculation or directly from a table. If there is doubt as to the 
applicability of one series, he may design and draw the two extreme 
sizes and the middle size, determine the applicability of any chosen 
geometrical series, and proceed by simple interpolation to tabulate 
dimensions for all sizes. A certain amount of caution is necessary 
in such proceedings and it is always best to draw at least the two 
extremes if one is in doubt as to the applicability of a given series. 
Hoffman points out a case in which certain handwheels were under 
consideration. A satisfactory result was obtained by grading the 
outer diameter according to the ^\/io, the thickness of the rim and 
spokes according to the \/10, and the diameter of the hub accord- 
ing to the (\/10) 3 - This looks extremely complicated, but if 
one draws the largest and smallest handwheels, plots the points 
on semi-logarithmic paper and draws straight lines between, the 
job is finished. 

German authors have been quite enthusiastic over the fact that 
geometrical series, and particularly geometrical series with ratios 
equal to roots of 10, have proved widely applicable to design and 
sizing as now carried out. They seem to emphasize the ease with 
which the series based on \/10 can be fitted into existing designs. 
At first sight it seems as though such applicability represented a 
most remarkable coincidence, but there is really much behind the 
phenomenon. 

In the first place, human beings accept geometrical ratios in 
preference to arithmetical when the results are presented in such 
fashion that the mathematical construction is not in evidence. 
This is a phenomenon well known to psychologists and one which 
has been extensively tested. Examples could be cited from many 
different human activities, but consideration of size is sufficient 
for the present purpose. The normal human being, in developing a 
series of sizes, starts out with small increments, enlarging them as 
he increases the sizes. He thus unconsciously approximates some 
geometrical series or some combination of geometrical series. 

In the second place, one can obtain practical approximations 
to all the numbers there are by using a sufficiently great number 
of series based on roots of ten. 

The preference for the series with the \/10 ratio is probably 
due to the large number of terms in the series, and to the binary 
structure of the number 80, which contains 2 as a factor four times. 



32 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

This last feature makes this series fit in with many existing sizes 
since, as has been pointed out, the binary principle (successive 
division or multiplication by 2) has been extensively used in frac- 
tional and in many other series of "preferred numbers" which have 
been developed and used more or less unconsciously through a long 
period of time. This feature also makes it possible, if changing 
industrial conditions make such a development desirable, to double 
the number of terms in a 5-, a 10-, a 20-, or a 40-series, or in any 
part of such a series. Furthermore, it may happen that this series 
represents approximately the path chosen by an uninfluenced hu- 
man being and is therefore mathematically located in accordance 
with the theory of probability. 

In presenting this paper the authors have attempted to picture 
the present status of the preferred-number idea and to do it in such 
a way as to point out its advantages and also its complications and 
dangers. The authors themselves hold no brief for preferred num- 
bers, but they do believe that the idea indicates possibilities of sim- 
plification and elimination of waste of such magnitude that thorough 
investigation is justified. 

DISCUSSION 

BUCKNER SPEED. 1 Most mechanical operations are cuttings or 
dividings; on the other hand, the system of counting as distinguished 
from cutting originated in the piling up of things one by one, as for 
example, money, soldiers, goods. 

For arithmetical operations, such as enumeration of population, 
or matters relating to money or goods, the decimal system is at every 
one's finger tips, deeply rooted in the very structure of our hands. 

For mechanical operations involving cutting in its broadest sense, 
the use of the scale divisible by two is equally old as the sword arm 
of our fair-haired ancestors. 

This is what is forcing itself on the users of the metric system, 
which is a beautiful doctrinaire product of the French Revolution 
and is useful in many cases, certainly in money counting, in certain 
kinds of buying and selling, and in some engineering and many other 
calculations. 

Now this preferred-number system is an endeavor to adapt the 
dividing-by-two system to the metric system. 

The chief purpose of a "size" is to enable the desirer of a certain 
size to most easily make his want exactly known to a supplier who 
has a number of "sizes" in his possession. The desirer would 
like to be able to find a supplier who had a stock graded by minutely 
small steps, but the supplier must in all reason only make and keep 
the fewest number of sizes that will retain his continued good-will 
with the desirer. 



Charge Spec. Studies, Western Elec. Co., New York, N. Y. 



DISCUSSION 33 

This, and this only, seems to me is the right rule for the grading 
of things into sizes. 

Not only is a rule such as the preferred-number system undesir- 
able, but it often happens that neither a simple equal-increment 
rule as the number sizes of hats, nor a double-the-volume rule as 
in bottles, nor a three-sizes-up-and-halve-the-area rule of the 
Brown & Sharpe copper wire gage will do after the demand exceeds 
the supplier's wish to restrict the number of sizes kept in stock to 
any set rule. 

There are also good size systems in which the magnitude of the 
intervals diminishes as the size of maximum demand is'approached 
and the intervals become greater among the rarely-called-for sizes 

These are matters of extreme practicality. 

The argument of saving in cost of design misses the whole point, 
for after all the chief use of a number or a letter designating a size 
is to afford a memonic key by which a person may make known to 
another person his desire, and not solely to fall into some general 
system. 

A. L. DE LiEEuw. 2 The troubles with past attempts at standard- 
ization have been twofold: either people did not do anything at 
all, or else they tried to go too far. 

In the paper under discussion a thing is brought out which seems 
to be getting around the standardization problem quite nicely. 
The part of the paper which interested me was about the preferred 
system of figures, and particularly the word "preferred," which 
means that we can use it but do not have to. Whatever system of 
standards we may want to apply, we must take care that we let 
nature take its course, and many attempts at standardization have 
failed because the course of nature and its vagaries have been dis- 
regarded. As soon as we make a standardization system ironclad, 
a thing which has no outlet for human tastes, vagaries, and foolish- 
nesses as soon as we do this, the thing falls by its own weight. 

That it is dangerous to lay down a system of standardization for 
future generations is vividly brought out by the present war between 
adherents and opponents of our English system of weights and mea- 
sures. Of course we had to have a system, at least in this case, but 
once having it we are tied to it and cannot let go. I do not mean 
to say that the English system is so perfect that it could not be im- 
proved, but I do wish to say that we cannot let it go, because we 
have it. 

Again I say I am not speaking of the advantages or disadvantages 
of any system of weights and measures. I am not claiming that 
the English system is better/or worse than the metric. I am speak- 
ing of this fact: that, having the English system, we cannot get the 
metric, whether we want it or not. We are tied by our standards. 
We cannot very well get away from such a thing as a standard of 



Cons. Engr., 149 Broadway, New York, N. Y. Mem. A.S.M.E. 



34 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

weights and measures, though this does not mean that we will have 
the same disadvantage when we standardize in other respects. 

The system of preferred numbers has, at least in its name, this 
element of broadness: that we have a system of standardization 
but are not compelled to follow it. 

Whether the proposed way of getting at the numbers is exactly 
right or not does not interest me very much, at least not at the 
present time. I believe, as the authors said, that it is merely an 
attempt to get the thoughts of the country, to get us to think about 
the matter,, and to decide whether this is the best way or not, and if 
not, to get something better. 

W. H. TiMBiE. 3 It would seem that the system of preferred 
numbers as presented in the paper could be bettered in one respect 
by a very simple change. It will be noted that in Table 4 there 
are several sizes numbered 1 and several numbered 2, etc. It seems 
to me that there would be a big advantage gained if in each system 
there were. but one number 1, number 2, and so on. For instance, 
according to the scheme as presented a glass numbered 2 could be 
any one of four sizes. I would suggest as a numbering scheme that 
the thousandths of the logarithm be used as the number. Thus, 
in the first column of the table we should have as numbers 0, 12Y 2 
25, 50, and so on. The size number 200 in one column would be 
number 200 in any column, regardless of what series it was tabu- 
lated in. A system that has such a scientific method of sizing 
should certainly have a scientific method of numbering the sizes. 

F. W. GURNET. 4 The thought that strikes me quite forcibly 
is how purely academic this whole discussion is. Of course there 
is no denying the fact that our various standards of sizes are far 
from ideal. The same thing may with equal force be said of any 
of the customs or laws or fashions or habits of the human race. 
What a tremendous nuisance, for example, is the Babel of languages 
that we so haltingly speak and that we cannot speak. But how 
completely we realize the utter impossibility of bringing all our 
discordant tongues into one harmonious language. Yet this is 
not more futile than to think of changing over the sizes of our nails, 
our frying pans, our bolts, or our shafting. Ideally, it might be 
desirable. Practically, it is impossible. To put across such a 
change would entail an expense of billions, and the confusion that 
would result during the process of transition would be almost be- 
yond estimate. And then it is very doubtful if these seemingly 
ideal standards when put into practice would prove to be as feasible 
as the present imperfect standards that have been worked out by 
the experience of generations. 

The whole matter is very like the movement for the adoption of 



* Prof. Elec. Engrg., Mass. Inst. of Technology, Cambridge, Mass. Mem. 
A.S.M.E. 

4 Ch. Engr., Gurney Ball Bearing Co., Jamestown, N. Y. Mem. A.S.M.E. 



DISCUSSION 35 

the metric system. It is susceptible of beautiful argumentation 
but is something which American industry won't stand for. 

E. A. JOHNSTON. 5 In a general way I am fully in accord with 
the theory or principle involved, but feel sure that the application 
will be difficult. While it is too true that our present sizes, num- 
bers, capacities, and gages are inconsistent and mean very little, 
I fear that it would be a stupendous undertaking to change these 
as applied to 'commodities already in common use. 

GEORGE S. CASE. 6 The paper shows in a very interesting way 
how size standards automatically fall into geometrical progression. 
This would be even more convincing if the authors made the 
best of their case. Wire nails, for instance, have for their most 
important factor surface area, which is proportional to the product 
of the length and diameter. Fig. 1 looks as if this would work out 
very close to geometrical progression. 

The case in Fig. 10 would be even stronger if the little-used 
dimensions were left out Vie, l 3 /s, 1 5 A, l 7 / 8 and 2 3 /4-in. bolts 
have almost entirely dropped out of use and would not be stocked 
in one per cent of the volume of other sizes. If these sizes are 
eliminated, the curves in Fig. 10 become almost perfect straight 
lines. 

It has been customary where sizes have been worked out scientifi- 
cally to use geometrical progression and it is hard to see that any- 
thing has been gained by the change of nomenclature to "preferred 
numbers." 

As an example of the possibility of standardization along this 
line, the company with which the writer is connected formerly 
used twenty-eight sizes of pasteboard cartons for packages of bolts. 
By setting up a new series of sizes, each one 15 per cent greater 
in capacity than the next one to it, it was found possible to cut in 
two the number of sizes and have the boxes more uniform than in 
the worst case under the old series. 

F. R. STILL. T In connection with the subject of size standard- 
ization by preferred numbers, it may be of interest to know that 
the steam engines we make practically fit in with this scheme very 
nicely. Taking the horsepowers as follows: 

10, 16, 25, 40, 64, and 100 

our sizes of engines with corresponding horsepowers would be: 

4 x 4, 5 x 5, 6 x 6, 7 x 7, 9 x 8, and 8 x 8 Double Cylinder. 

CARL G. EARTH. Being somewhat musically inclined I have 

looked upon the geometrical progression of an adjusted musical 

scale as a fundamental one handed us by nature herself. In this 



6 International Harvester Co., Chicago, 111. 

6 Lamson & Sessions Co., Cleveland, Ohio. 

7 Vice-President and Secretary, American Blower Co., Detroit, Mich. 
Mem. A.S.M.E. 

8 10 S. 18th St., Philadelphia, Pa. Life Mem. A.S.M.E. 



36 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

the simple number 2 is the basis as against 10 in the progressions 
presented in this paper as having been proposed for adoption on 
a large scale in Germany. In the musical scale the "interval" 
between two successive semitones is \/2~= 1.05947, i.e., the ratio 
of their respective number of vibrations per second is 1.05947. 

So long as we here in America stand together with England in 
refusing to adopt the metric system of weights and measures, I 
think we had better also adopt a system of geometrical series 
with 2 as the basic number rather than 10. Referring again to the 
musical scale, the vibrations of two successive octaves therein 
have this simple ratio 2. 

Some ten years or so ago, in making a slide rule for the quick 
determination of the strength and deflection of coiled springs for 
one of my clients, I discovered, as pointed out in this paper, that 
the Brown & Sharpe wire gage is based on a geometrical progression. 
To my chagrin, however, I found that it seemed void of a simple 
ratio anywhere, though 2 came as close to being such a number 
that it seemed a shame that its constant ratio had not been made 
V/2 = 1.1264, instead of 1, as I found it to be after getting in 
touch with Mr. Burlingame about the matter. 

I will also mention that for more than 15 years I have tried to 
persuade, and mostly succeeded in persuading, my clients to adopt 
standards of hour rates for both machines and men that approxi- 
mately follow geometrical progressions. Along with this effort 
I have also insisted upon eliminating any rates with fractional cents. 
A scale of hourly rate of this kind is therefore finally composed of a 
number of short arithmetical progressions pieced together so as to 
form a geometrical progression only in the averages of its arith- 
metical fields. 

LUTHER D. BURLINGAME. 9 A discussion under the title of 
Preferred Numbers does not to my mind convey the principal and 
underlying thought which is most important to be brought out, 
namely, that a series in geometrical progression will for many 
purposes give the least number of sizes and at the same time meet 
the needs more fully than any other possible series of sizes. This 
has nothing necessarily to do with preferred numbers and this 
principle has been applied in practical ways for many years, al- 
though not as fully as might be desirable. 

Such a principle was made the basis of the sizes of the American 
Wire Gage which was brought out by Brown & Sharpe Mfg. Co. 
in about 1857 and which provides for the diameter of wire following 
a geometrical progression, so that because of being in geometrical 
progression the steps between the small sizes are much less than 
between large sizes, as practical requirements dictate. 

The application of this principle to the feeds of Brown & Sharpe 
milling machines was adopted by this company in the 80's, and 

9 Industr. Supt., Brown & Sharpe Mfg. Co., Providence, R. I., Mem. 
A.S.M.E. 



DISCUSSION 



37 



in 1894 Mr. Carl Earth filed an application for a patent in which 
he emphasized the importance of a geometrical progression for 
geared feeds. 

It is interesting to find how closely good practice has followed 
geometrical progressions in mechanical design, even though such 
designs were not originally laid out with that thought definitely 
in view. An example is in milling-machine proportions where 
the lengths of table traverse as agreed on by the manufacturers 
are in approximate geometrical progression from the smallest to 
the largest machine. 

Another example of this is in the proportioning of taper shanks, 




FIG. 21 DIAGRAM SHOWING VARIATIONS OF DIAMETERS OF BROWN & 

SHARPS, MORSE, AND JARNO TAPERS FROM A TRUE GEOMETRICAL 

PROGRESSION OF SIZES 

regarding the standardization of which there has been much recent 
discussion. The accompanying diagram, Fig. 21, shows how 
closely the B. & S. tapers and Morse taper follow a geometrical 
progression. 

The "Jarno" taper, on the other hand, varying by tenths of an 
inch from size to size, gives as large a step between the small sizes 
as between the large, and when carried up to 14 in. in diameter, 
as in the case of the B. & S. standard, would require about 140 
sizes of the Jarno taper, an impractical series which would require 
a selection of sizes for actual use. The small unit of variation in 
diameter between the sizes of the Jarno taper, in this case 0.1 



61315 



38 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

in., makes this illustration comparable with the need for preferred 
numbers for metric units where so small a unit as the millimeter 
is used as the basis of the system for mechanical work, thus ex- 
pressing even moderate sizes in very large numbers. In such a 
case there is a great deal more need for a system of preferred num- 
bers than in the" case of our convenient system with the inch as 
the usual unit for mechanical work. 

This objection to large numbers is pointed out by the authors, 
where they show the objection to using a measurement of 324 in., 
for example, instead of expressing the value in feet and yards. 

At a recent meeting of the A.S.M.E. Committee on Plain Limit 
Gages for General Engineering Work the question of the steps 
where a change in limits and tolerances should be made in shafting 
was under consideration, and it seemed to be very natural to de- 
termine on a geometrical progression from 1 / 2 in. to 8 in. with 2 as 
a multiple, thus making the sizes 1 /2, 1, 2, 4, and 8. In this case 
the metric equivalents would be approximately 13 mm., 25 mm., 
51 mm., and 203 mm., so that rather than to use these odd numbers, 
selections could be made from a system of preferred numbers which 
would be approximations of the sizes desired. 

If an attempt should be made to apply such numbers to the inch 
systems a range which would fit to this system would not be adapted 
to other needs say, a system of machine keys. 

Even with the metric system, where the need of preferred numbers 
is accumulative, there is difficulty in making the same series apply 
to varying conditions. Thus, much standardization along mechan- 
ical lines already adopted in Germany would be changed if brought 
in line with their proposed series of preferred numbers. 

L. B. TucKERMAN. 10 Standardizations always mean on the 
average a use of material in excess of actual need in order to save 
the excessive labor cost of fitting the material most sparingly to 
each particular use. With the recent increase of labor costs the 
range of economic standardization of practice has been widely 
extended, but whether this range has been sufficiently extended 
to include the sweeping standardization of a preferred-number 
system, is a very debatable question. 

The standardization involved in a preferred-number system is 
so drastic and so complete that it should only be adopted after 
the fullest discussion of all the problems involved, including in 
particular a thorough study of all possible principal series, and all 
laws for the derivation of secondary and tertiary series. The need 
for such a thorough preliminary study of a proposed standardiza- 
tion is the greater the wider the scope of the proposed standards. 
It is also well known that it is easier to introduce some kind of 
standardization into a wholly chaotic practice than it is to introduce 
new standards into an already standardized practice. 



19 Engineer Physicist, Bureau of Standards, Washington, D. C. 



DISCUSSION 39 

The more thorough the standardization the more difficult be- 
comes any change or improvement, and a preferfed-number system, 
embedded as it would be in all the standard practices of the nation's 
industry, would offer great obstacles to any change whatever. 
It is therefore of the greatest importance that a preferred-number 
system be adopted only when it has been shown that relative costs 
of material and labor justify its adoption, and even then only when 
it seems certain that the most advantageous series has been selected. 
A poorly chosen preferred-number system, although it might result 
in a considerable temporary advantage in furthering standardiza- 
tion where standardization is needed, might later become a detri- 
ment due to the great difficulty involved in replacing it by a superior 
system. 

The thing which characterizes a preferred-number system over 
all other standardizations is its universality. It is intended to be 
applied to industrial products of the most diverse kinds, windows 
and automobile wheels, letter paper and tin cans, and especially 
to the innumerable products of automatic machine tools. 

The immediate application to all of these is, of course, not in- 
tended, but the ultimate value of the system lies in the gradual 
absorption of all standardized articles into the one system of pre- 
ferred numbers. 

For this reason it is necessary that the preferred numbers be 
chosen on a basis as nearly universal as possible, and that it contain 
within itself all the necessary flexibility to adapt itself to the most 
widely differing applications. 

A cursory examination of standard sizes of commercial materials 
such as had been made by Messrs. Hirshfeld and Berry im- 
mediately indicates the geometrical series as the only type of a 
series which over wide ranges of sizes will approximately represent 
the needs of a series of standardized products. The larger the 
number of examples collected, the more strongly is this conclusion 
borne out. 

There can be no question, then, that a preferred-number system 
must be based on a geometrical series. The problem of the most 
suitable geometrical series to be chosen as the basis of the preferred- 
number system is not simple. The application of the system is so 
wide that very diverse considerations govern different applications, 
and the choice must finally represent a compromise in which the 
relative importance of many conflicting needs has been carefully 
weighed. 

In a geometric series such as a n = cw" it seems obvious that 
there is no advantage to be gained in choosing any other number 
than 1 as the first term of the series. The series then would have 
the form o = r n . 

In the choice of the ratio, r, the series should contain only a 
finite and not too large number of different incommensurate ratios. 
r should therefore be of the form, r = p 1/a = \/pi" where the 



40 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

base of the system (p) and the exponent of the system (q) are 
relatively simple integers. Messrs. Hirshfeld and Berry discuss 
such values as 

Vio", Via Vio, 'Vio", etc. 

As our number system is a decimal system there is an obvious ad- 
vantage in making the base of the system 10 (or perhaps 100), 
since then the digits of the preferred numbers repeat in each decade 
(or century) of higher order. So obvious is this advantage that 
practically all of the series so far proposed for a preferred-number 
system have 10 as a base. From certain other considerations it 
would seem that 12 might be a preferable base, but with the decimal- 
number system so firmly established that no change in it can be 
reasonably contemplated, all these advantages are outweighed by 
the obvious advantages of 10. 

The full advantage of 10 as a base can of course only be secured 
by the use of a decimal system of units, and it is difficult to see how 
a preferred-number system can be made practically workable in 
the English system of units. It seems fairly clear that the adoption 
of a universally applicable preferred-number system as a basis for 
standardization presupposes for its successful working the adoption 
of a system of units with a common ratio of units such as the 
metric system, so that the base of the preferred-number system 
may be chosen the same as the standard ratio of units. 

There is no such obvious indication for determining the proper 
exponent of the system as governs the choice of the base. Shall 
the primary series be a coarse series from which secondary and 
tertiary series are obtained by interpolating finer steps? Or shall 
the primary series be a fine series from which secondary and tertiary 
series are obtained by skipping steps in the primary series? 

There seems to be no clear, unambiguous answer to either of 
these questions. The Germans have adopted 10 as the exponent 
of their primary series, i.e., n = \/10 thus choosing a coarse 
primary series and obtaining their finer secondary and tertiary 
series by dichotomy, 

rz = Vio, n = Vio, n = Vio, etc. 

The only reasons given for this choice are that it seems to allow 
sufficient flexibility for the few illustrative examples examined and 
that it fits in well with a series of standard diameters (DINorm 3) 
already adopted by the Normenausschuss der Deutschen Industrie. 
These reasons are sufficient to establish a temporary advantage in 
the adoption of this system, but it by no means insures that the 
choice will not prove burdensome later as the field of application of 
preferred numbers widens. 

There are very definite reasons for doubting whether this choice 
is a wise one and whether in particular the principle of dichotomy 



DISCUSSION 41 

is the proper principle to be used in interpolating a fine series into 
a coarse series of preferred numbers. 11 

It is seldom that a standard series of articles are geometrically 
similar because of the fact that it is seldom that they are 'con- 
structed with only one limiting physical requirement. This is 
illustrated by the series of handwheels discussed in detail by Hbf- 
mann in his report to the Normanausschuss in which different 
dimensions are graded according to the different series \/10, 
\/10, and \/ r \Q s . (Incidentally this last gradation also shows 
that they have already found the fundamental principle of their 
series insufficient for all cases, in that they have introduced 
a new coarse series \/10 3 by skipping terms in a fine series.) 
Illustrations of this fundamental fact can be multiplied indefinitely. 
Nails are varied in length and the diameter simultaneously varied 
(as noted by Hirshfeld and Berry) to provide adequate strength. 
Pipes are varied in diameter to provide for larger flow and at the 
same time are varied in thickness so as to stand the pressure. 

It seems impossible to state any general law of such double, 
triple, or multiple requirements on the dimensions of standardized 
articles, but in an overwhelming majority of cases the requirements 
are approximately expressed over a wide range by the relations 
(AiLi) B - = (k,L 2 ) a - = (k*L 3 y* = (W 4 = etc. Here L t , L 2 , 
L 3 , etc., are different dimensions of the article, fci, k 2 , k 3 , etc., nu- 
merical constants, and the exponents ai, a 2 , a s , etc., are in general 
simple numbers, 1, 2, 3, 4, 5, etc., rarely as large as 5. Thus 
in the example of Hofmann's handwheel (&]Li) 4 = (& 2 L 2 ) 2 = (A^Ls) 3 
where LI is the outside diameter, L 2 the thickness of the rim or 
the spokes, and L 3 the height or diameter of the hub. Dr. 
Buckingham has calculated an interesting case of a ball bearing 
designed to support an overhung flywheel. Here he finds the rela- 
tion (fcjLi) 4 = (& 2 L 2 ) 5 [where LI is the diameter of the balls and L 2 
the diameter of the shaft] is necessary to insure proper stresses in 
the shaft. 

These relationships suggest the desirability of making the pri- 
mary series a fine series and making the exponent q divisible by as 
many small integers as possible. 60 suggests itself as a possible 
choice. Making \/10 the primary ratio, a number of coarse 
secondary and tertiary series can be found by taking every second, 
third, fourth, fifth, sixth, tenth, twelfth, fifteenth, twentieth, or 
thirtieth term in the original series. This would include the pri- 
mary (\/10) and secondary (\/10) series of the present German 
proposal, and would avoid the awkwardness of such ratios as \^W. 

It is not intended here to propose definitely that \/10 should be 

11 The writer is indebted to Dr. E. Buckingham, of the Bureau of Stand- 
ards, for assistance in formulating these ideas and in particular for the loan 
of a manuscript discussing in a fundamental way the principles underlying 
the choice of a preferred-number system. 



42, SIZE STANDARDIZATION BY PREFERRED NUMBERS 

made the primary ratio of the series, but merely to indicate some 
of the considerations which should be taken into account before a 
preferred-number system is adopted. It may very well be that 
there are advantages in a dichotomous derivation of secondary and 
tertiary series which would outweigh the presence of 3 as a factor 
in the exponent of the series, but the fact that 3 occurs so frequently 
as an exponent in physical laws certainly indicates that its advan- 
tages and disadvantages should be carefully weighed in selecting 
a preferred-number system. 

It is the practice in Germany to round off the preferred numbers 
to the nearest unit or, where that gives too great a discrepancy, 
to the nearest tenth. In some discussions it is implied that this 
rounding off is merely a concession to a natural dislike of complicated 
figures. A closer examination shows that the demand for rounding 
off rests upon a much firmer basis. The dimensions of articles are 
conditioned not only by the requirements which each individually 
must meet, but also in large measure by their mutual relationships. 

Fundamental in these relationships is the need of fitting. So 
fundamental is this need that in many lines of manufacture whole 
series of special fittings are manufactured to meet it. A simple 
illustration is the series of reducing L's, T's, and couplings used in 
fitting pipe of different sizes. Here the very nature of the objects 
demands specially adapted fittings. In many lines, however, 
proper dimensioning of standard sizes is all that is necessary. In 
the common brick, for instance, the breadth and length are so 
chosen that the header courses fit with the stretcher courses without 
the use of bats as fillers. 

The basic requirement of fitting of different standard sizes with- 
out special adapters or laborious shaping at fitting points, is that 
the fitting dimensions shall have simple commensurate ratios. In 
the rounded series of preferred numbers this requirement is met 
as far as possible, without too wide departure from the basic geo- 
metrical series. The rounded values of the German primary 
series, VlO 1, 1.2, 1.6, 2, 2.5, 3, 4, 6, 8, 10 have a maximum 
discrepancy of about 5 per cent from the geometric series at 3 and 
6. Whether this discrepancy is too great a price to pay for their 
simple fitting possibilities can only be decided by careful investi- 
gation. It would be worth while to find out whether any different 
coarse series would permit of as simple rounding off without greater 
error. 

In many fields of application this problem of fitting without 
special fittings does not of course enter. In any round stock, 
such as pipes, shafts,. wheels, etc., special fittings are always in the 
nature of the case necessary. 

This raise's the question whether a single preferred-number series 
should be used, both for those cases where fitting is an important 
problem and for those cases where it is not required, It may 
very well be that hot one but two or three , or perhaps more rad- 



DISCUSSION 43 

ically different preferred-number systems might ultimately be more 
advantageous. 

E. R. JlEDKiCK. 12 The reason for the selection of the relation 
r = \/10 is n t always stated clearly. Any system of preferred 
numbers that did not include 10 as a ratio between two of the num- 
bers would be open to serious question. But 2 is also a convenient 
multiple. It seems to me that the well-known fact that 2 10 = 1024 
is the essential reason for the choice of the preceding ratio, since 
10 3 ( = 1000) is so very nearly equal to 2 10 . Whence we have 
10 3 = 2 10 (approx.) or VlO =_v / 2 (approx.). For this reason 
the system based upon r = \/10 will also contain multiples of 2 
to within the degree of accuracy commonly accepted in the process 
of rounding off. While it may be argued that the numbers 2, 4, 
do not occur among the preferred numbers, it has been pointed out 
repeatedly by many that the decimal point may be shifted in the 
preferred numbers as stated. After this is done it will be seen 
that the numbers 2, 4, 8, 16, 32, do occur, as also the numbers 
0.5, 0.25, 0.125. If it were not for the rounding-off process the 
multiples of 2 would continue to appear. We cannot hope that 
they will continue forever in any rounded scheme, but it would be 
desirable to retain 64, which is under debate, if we can. 

Finally, I may remark that any two geometrical progressions 
that both contain the same two numbers must also show other 
numbers in common. This remark becomes important in con- 
nection with the figures shown in the paper by Messrs. Hirshfeld 
and Berry. In that paper many of the figures are drawn on semi- 
logarithmic paper. If we set S = Ar", where S is the size, r the 
fixed ratio, and n the size number, then log S = log A + n log r. 
If we plot log S against n on ordinary paper or S and n against each 
other on semi-logarithmic paper, the figure will be a straight line 
and its slope will be log r, i.e., Vio, if the scales are the same in, the 
two directions and if logarithms of the base 10 are used with r = 
\/10. Since this would be a very inconvenient slope, it is prefer- 
able to take the units in the direction representing size numbers 
very much smaller. In this paper the units seem to be in about 
the ratio of one to ten, so that the line appears to have a slope 
of about one, that is to say, it makes an angle of about 45 deg. 

The connection of this with my previous remarks is that the slope 
will be log r in any event, but the same straight line can be used for 
different values of r by changing the unit in the direction that 
represents size numbers. This would mean, for example, that ^ the 
vertical lines in any of the figures may be omitted entirely and ; aiiy 
desired number of new vertical lines can be inserted in their stead, 
provided only that the first and last lines be kept in place and that 
the new verticals be equally spaced. For the effect of doing :this 

12 Professor of Mathematics, University of Missouri, Columbia, 'Mo. 
Mem. A.S,M>,E. ...-,". 



44 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

is to change the scale in the direction that represents the size 
numbers. It seems to me that it is very important for those who 
use these figures to understand this that the slope of such graphs 
is not very significant. Otherwise they may attach too much 
significance to the apparent slope of these lines, and to the differ- 
ences in the slopes in two different figures. 

In connection with the application of preferred numbers to sizes 
of cartons, boxes, etc., if the controlling linear dimension would 
follow the preferred-number series, the volumes would follow a 
geometrical progression with the ratio (-\/10) 3 which is approxi- 
mately equal to 2. The effect is the same as choosing for the vol- 
umes every third number of the preferred-number series, thus 2, 4, 
8, 16, 32, 64 (according to the German series). As before noted, 
1, Vz, 1 /4, Vs also occur in the preferred-number list if the decimal 
point is moved in the customary manner. It should be noticed 
that intermediate sizes for such containers should be inserted 
two at a time, since doing so would again result in a geometrical 
progression whose ratio is \/10. 

The size series for druggists' bottles is an interesting example. 
Thus the present sizes are 1, 2, 4, 8, 10, 12, 16 oz. The only de- 
viation from the regular preferred-number series being in the case 
of the 12-oz. bottle which would become 12.5 oz., since 10 and 12.5 
are the preferred numbers between 8 and 16. From the present 
standpoint, these would count as intermediate sizes in the main 
series 1, 2, 4, 8, 16. 

A. E. KENNELLY. 13 The system of selecting sizes described in 
the paper would cause such widespread and radical changes in 
production that we should regard the system at present as a theory 
of economic design for future gradual introduction, rather than for 
immediate adoption. The system would be a much more radical 
innovation than the metric system of weights and measures. In 
order to introduce the metric system into production we should not 
have to change any size, but only the numerical values of the di- 
mensions of existing things. On the other hand, to adopt a system 
of preferred new sizes in geometrical ratio for hats, shoes, frying 
pans, window panes, pipes, etc., would probably mean changing or 
rejecting machines and machine parts, in many cases at great 
expense. 

Probably a geometrical series of preferred numbers would permit 
of realizing the minimum number of commercial sizes and of stand- 
ardizing such sizes in the simplest and most scientific way. The 
80th-root-of-10 series, with its constant ratio of 1.0292 = log" 1 
0.01250, or a nearly 3 per cent step, seems to offer, as suggested 
in the paper, a very sound basis of advance; because a 3 per cent 
step is probably sufficiently small to provide for fine gradations 
where needed, and skipping intervals in the series systematically 

Prof, of Elec. Engrg., Harvard University, Cambridge, Mass. 



DISCUSSION 45 

it gives successively by inclusion the 40th-root series of about 6 
per cent (1.0593 = log" 1 0.0250) the 20th-root series of about 12 
per cent (1.122 = log- 1 0.0500), the lOth-root series of about 26 
per cent (1.259 = log" 1 0.100) and the 5th-root series of about 60 
per cent (1.585 = log" 1 0.200). In all such cases the new series is 
most readily denned by the logarithm; thus the preceding sizes 
would be naturally designated as sizes 25, 50, 100 and 200. 

From a purely theoretical point of view, it is a matter for dis- 
cussion whether the basis of the geometrical ratio should be a 
root of 2 or a root of 10. In other words, some would prefer 
that the series of steps should pass exactly through the binary powers 
2, 4, 8, 16, etc., whereas others would prefer that the series should 
pass exactly through the decimal powers 10, 100, 1000, etc. The 
recommendations of the paper are in favor of the decimal system. 
This is probably the better choice, from the standpoint of arith- 
metical computations and of engineering design, especially when it is 
seen that the decimal steps pass very nearly through the binary 
powers (1.995 = log- 1 0.300, 3.981 = log- 1 0.600, 7.943 = log- 1 
0.900, etc.). 

G. M. EATON. 14 From the standpoint of adaptability of the 
preferred-number principle to the electrical manufacturing industry, 
we may divide the output into two major classes: 

1 Apparatus limited only fundamental natural law 

2 Apparatus subject to rigid physical limitations of an 
arbitrary nature, as well as by natural law. 

The second class is well illustrated by axle-hung railway motors. 

There is nothing fundamental about the track gage, but in view 
of the overwhelming existent investment in track, motive power, 
and rolling stock tied up to present standards, the gage ranks with 
fundamental natural law as a limiting feature. 

If the railway industry were starting now untrammeled by 
precedent, some arbitrarily determined gage would necessarily be 
selected as the standard, and it certainly would not be our present 
standard. The motor ratings, car sizes, weights, etc., could in all 
probability follow the logarithmic law, though I am not yet sure 
whether this would be a wise arrangement. The ratio of active 
armature-core length to diameter could probably be fairly uniform 
for the smaller motors, being selected at a value that would give 
the best average weight and cost efficiency. 

As soon, however, as we pass a rating where this ratio results in 
a motor which with its gear fills the available space between the 
wheels, we must depart from this ratio, the diameter increasing 
while the length gradually decreases as the gear face and other 
parts demand more room. 

Graph 18,' Fig. 22, shows the present W. E. & M. Co. standard 



> Ch. M. E., Westinghouse Elec. & Mfg. Co., E Pittsburgh, Pa. Me 
A.S.M.E. 



40 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

d.c. railway motor horsepower per 100 r.p.m., while graph 19 shows 
armature diameters, and 20, the lengths of active core. 

Graphs 35, 36, and 37 show for the same motors the rating at 




GRAPHS: OF Hi>. PER 100 R.P.M., ARMATURE DIAMETERS, AND 
CTIVE CORE FOR WESTINGHOUSE ELEC. & MFG. Co. RAILWAY 

AND MiNE-LoCOMOTIVE MOTORS 



40 per cent of maximum r.p.m., and the corresponding core lengths 
and. diameters,. . 

The departures from the logarithmic characteristic form a very 



DISCUSSION 47 

long story of evolution, common in principle, no doubt, to many 
other lines of American products. 

In addition to the gage limitation, there is of course a long list 
of other features, such as clearance dimensions, etc., which in effect 
are fundamentally binding. 

A more fundamental limit is met in coal and other mining opera- 
tions. I refer to thickness of vein, either height or width as the 
case may be, which produces a profound effect on the proportions 
of apparatus built to serve the mining industry. 

Graphs 21, 22, and 23 (and 38, 39 and 40) show for W. E. & 
M. Co. mine-locomotive motors the features referred to for railway 
motors. 

Railroad clearances and other shipping conditions also affect 
other lines of apparatus in the larger sizes, imposing restrictions 
which modify proportions. Below these limits, however, very 
logical lines of ratings have been developed, some of which approxi- 
mate the logarithmic characteristic. 

Almost all of the graphs show a drooping tendency. This is 
most sharply outstanding in cartridge fuses. All these lines have 
been developed to meet the demands of the trade. We must there- 
fore satisfy ourselves as to the absence of sound reasons for a 
drooping tendency in the curves before we admit that a logarithmic 
characteristic would have produced better overall results or, in- 
deed, results equally good. 

Let us study the fundamental psychological and economic reason 
for the drooping tendency of the rating characteristic. 

Let us assume that we have a customer who wishes to purchase 
a 12-hp. motor, and that we have in the 5th-root series only a 10-hp. 
and a 16-hp. rating available. With headway established for the 
preferred-number principle and with good salesmanship there 
should be comparatively little difficulty in satisfying him with the 
larger motor. 

But when our next customer calls for a 120-hp. motor, it will 
be hard to show him that a 160-hp. motor is the right article. He 
will not be convinced by talk of ratios, because his mind will focus 
on the number of dollars extracted from his pocket to purchase a 
machine with a capacity in excess of his fundamental needs. 

I hear the rebuttal say that the purchaser of ten 16-hp. motors 
will have to face a larger dollar differential than the man who buys 
one 160-hp. motor, and that the demand for small motors is numeri- 
cally vastly greater than for large motors. The answer of course is 
obvious: viz., that in the final analysis we must study the re- 
action of the ultimate user, as the large orders for small machines 
are usually placed by manufacturers who assemble them in their 
own product, or by jobbers who resell them to the user. 

Of course the rebuttal will say that in a preferred-number world 
no customer will have any use for a 120-hp. motor, and I will grant 
this when railway grades, the thickness of coal seams, the depth of. 



48 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

ore bodies, logical hydraulic head in power developments, and all 
the other things that combine to make the earth what it is, are 
arranged on a preferred-number basis. 

In the case of the electric motor, a fundamental handicap is 
imposed on operating cost, as well as first cost, when an oversized 
motor is used. I refer to the bad power factor that is involved. 
This is a serious matter and would require most careful attention. 
It means that the application must be adapted to the motor. If 
this could always be done economically it would inevitably reduce 
the number of sizes, and ultimately save the people's money. 

It is vastly more important in motor manufacture to reduce 
the variety of frame sizes, compound dies, coil formers, jigs, tools, 
etc., than to limit the ratings, and for many years the closest 
attention has been devoted to this end in bringing out new lines 
of motors. I realize that a motor in the 10-series would closely 
fit the requirements of the case of our customer referred to. This 
at first thought comes close to simply calling our present practice 
a preferred-number system and explaining away all departures from 
the true logarithmic characteristic. 

But on closer analysis one cannot fail to be impressed with the 
remarkable coincidence of the characteristic curves of some im- 
portant lines of apparatus with the fifth- and tenth-root series. 

For example, graph 28 describing our type CS motors is very 
close to the fifth-root curve in the lower ranges, and is practically 
in exact coincidence with the tenth-root curve for the larger motors. 

This feature is in evidence in some of the other lines. The point 
has been emphasized in graph 28 and a few of the other graphs by 
adding light dotted lines parallel to the fifth- and tenth-root curves. 

We regard this as an important indication of the logic underlying 
the selection of these particular characteristics by the German 
Committee. 

We have plotted a good many other lines of our standard appara- 
tus and find such a prevalence of the drooping characteristic as 
to lead us to suspect that it represents a fundamental law, since 
the wide range of lines studied and the varying industries served 
by these lines should iron out individual tendencies. Later we 
find the tendency prevalent in the graphs presented in Messrs. 
Hirshfeld and Berry's paper. 

Going now a little further into the details of our product, we will 
consider motor r.p.m. We at once face fundamental speeds in 
synchronous a.c. motors. These speeds, on any fixed frequency of 
supply circuit, are an inverse function of the number of poles, 
and the poles are in even numbers. The speeds therefore funda- 
mentally lie in an arithmetical series, as the steps would be too wide 
if we limited the number of poles to a geometrical series of 2, 4, 8, 
16, etc. In our practice, we change the progression of the series 
at 32 poles, stepping by twos up to 32 and by fours above that 
figure. It is interesting to note that the trade conditions which 



DISCUSSION 49 

demand closer size gradations in the larger sizes can be met in the 
same sizes by wider speed variations. 

In industrial applications, trade conditions demand speed ratings 
for d.c. motors in line with the fundamental speeds referred to 
above. For example, a machine tool must be adaptable, without 
change of gear train, to either a.o. or d.c. power supply. Therefore, 
in electric-motor manufacture the vital factor of r.p.m. apparently 
cannot be influenced by any arbitrarily selected geometrical system. 

Our rotating machines in particular, and in a lesser degree our 
stationary apparatus, are built up about fundamental relationships 
that are of the same binding nature as the r.p.m. discussed above. 

E. H. RiGG. 15 The system of preferred numbers will be sub- 
jected to considerable investigation before being formally accepted 
by, for instance, the manufacturers of structural steel rolled to 
shape, such as channels, etc. Rolled steel is a subject which comes 
at once into the mind of a shipbuilder; it is one on which a vast 
amount of standardization work has already been done, and its 
total use in America is tremendous. 

Taking American structural channel sections as an instance, we 
are accustomed to a series in which the depth of web in inches 
increases as follows: 3, 4, 5, 6, 7, 8, 9, 10, 12 and 15. The ten sizes 
constituting this series have been plotted in Fig. 23. It will be 
noted that the depths of web, weights, and section moduli do not 
run either in fair curves or straight lines. 

The same thing has been done for British standard channels 
(narrow flanges) for the purpose of comparison and greater lack 
' of fairness in the curves shows up (see Fig. 24). The ultimate 
requirement is strength, which is represented best by the section- 
modulus curve. It is a question worthy of further study whether 
a more economical set of channels could not be designed having 
section moduli varying somewhat as indicated by the dotted straight 
line joining the present minimum and maximum. The other 
features would vary approximately as the other dotted straight 
lines. It should be understood that these dotted lines have, for 
present approximate purposes, been drawn in by arbitrarily joining 
minima and maxima. 

It will be noted that the straight lines applied to the American 
standards come considerably nearer to the mean of the curves 
than in the case of the British standards. 

Varying thicknesses of each size will still be needed in order to 
get a finer grading of section modulus than would be obtainable 
otherwise. The possibilities are indicated by the dotted lines on 
Fig. 23. It will probably be found that varying the thickness of 
each full size will be more economical commercially than the 
introduction of half-sizes, but this point is worthy of further con- 
sideration. There has been some discussion looking to the adoption 

New York Shipbuilding Co., Camden, N. J. 



50 



SIZE STANDARDIZATION BY PREFERRED NUMBERS 



of a single American standard line of channels for structural and 
ship work. An analysis along the lines of the preferred-number 
system would be in order and should by all means be made prior 
to the adoption of any economical single standard. In comparing 



Li/rye of Section 

Modulus divided b 

Yfeiqh+(MinimaJ 




I 2 .3 4 5 6 7 89 
Scale of Sfjes. 



FIG. 23 AMERICAN STANDARD STRUCTURAL CHANNEL SECTIONS CURVES 
FOR DEPTHS OF WEB, WEIGHTS, AND SECTION MODULI 

the American standards (Fig. 23) with the British (Fig. 24), the 
greater influence of the shipbuilding demand in Great Britain must 
be kept in mind. Flange width and flange thickness in shipbuilding 
have to meet conditions different from those in buildings and 
bridges, and herein lie the difficulties of a single standard for 
America. 



DISCUSSION 



51 



It is obvious that a full study of channel sections alone would be 
a decidedly lengthy affair, to say nothing of the other commercial 
rolled shapes. These remarks merely scratch the surface. 

WILLIAM A. DEL MAR. 16 The system of preferred numbers re- 



us about Axis 1-! (Inches 3 ) 
5 8 o S 2S S 






























poooo'o- 
IS o > oi <y b> o i 

ale of Depth of Web. in Inches, and Weiqht per Foot Run, in Pounds. 
to oi * oiov CP 5 g ooo 
























2 


























/ 


'/ 


























/' 


























| 


I 
























^ 






















f > 


\o? 
























1 


w 




' 














Scale of Section Modu 

io o ^ <yi c 










0, 

A/ 




/ / 






















G ></ 


/ 




























/ 












/ 












/ 


/ 












/ 


^ 










, 


9 












^ 


^ 












//1 










J 


^ 






^x 










^ 




X 


4 


^ 




X 





7* 












7 


> 


' 




t?^J 


^ 


^ 














j 


x x 


^ 


^ 


X 


" 










r_ 






// 





y< 


^^\ 










<& 








x 


',', 


X 


fcw 


-K of Section Modulu 
Divided by Weight 






/^ 


V 


s 








/x 





















































23456769 10 II 
Scale of Sijes 

FIG. 24 BRITISH STANDARD CHANNEL SECTIONS CURVES FOR DEPTHS 
OF WEB, WEIGHTS, AND SECTION MODULI 

cently introduced in Germany and France presents no novel fea- 
tures to the wire and cable industry of the United States, as such a 
system has been in use in that industry since 1857 when the Brown 
& Sharpe gage was introduced. 

The preferred numbers of the Germans are a series of numbers in 

16 Habirshaw Electric Cable Co., Yonkers, N. Y. 



52 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

geometrical progression, i.e., related to onejmother by a constant 
ratio having the values \/W, -\/10j or \/10, depending upon the 
series. The Brown & Sharpe gage, now known as the American 
Wire Gage, has the constant ratio \/92 between the diameters of 
successive sizes. This equals 1.1229 and \/10 = 1.1221, so that 
it is obvious at once that the two systems are substantially the 
same, the difference being less than 0.08 per cent. 

A comparison between the diameters of A.W.G. sizes and the 
secondary series of preferred numbers is shown in Table 6. As 
would be inferred from the similarity of the ratios noted above, the 
preferred numbers come very close to the A.W.G. diameters ex- 
pressed in mils to the same number of significant figures. Thirteen 
out of thirty-four sizes show the two systems in exact agreement, 
and the maximum divergence is 2% per cent. 

When it comes to the application of preferred numbers to stranded 

TABLE 6 COMPARISON OF A.W.G. SIZES WITH FRENCH AND GERMAN 

PREFERRED NUMBERS 

A.W.G. Diam., Preferred Numbers A.W.G. Diam., Preferred Numbers . 

No. mils. German French No. mils. German French 

30 10 10 10 13 72 72 72 

29 11 11.2 11.2 12 81 80 80 

28 13 12.5 12.5 11 91 90 90 

27 14 14 14 10 102 100 100 
26 16 16 16 9 114 112 112 
25 18 18 18 8 128 125 125 

24 20 20 20 7 144 140 140 
23 23 22.5 22.4 6 162 160 160 
22 25 25 25 5 182 " 180 180 

21 28 28 28 4 204 200 200 
20 32 32 32 3 229 225 224 
19 36 36 36 2 258 250 250 

18 40 40 40 1 289 280 280 
17 45 45 45 325 320 320 
16 51 50 50 00 365 360 360 

15 57 56 56 000 410 400 400 
14 64 64 63 0000 460 450 450 

copper conductors, certain difficulties stand in the way of using the 
system in any way except for the diameters, of the component wires, 
because the number of wires in a concentric cable must be according 
to the series 7, 19, 37, 61, 91, 127, etc. 

G. F. JENKS. 17 Many years ago an ordnance engineer proposed 
a series of guns advancing by geometrical series according to weights 
of projectiles or cubes of calibers. Although various countries have 
not standardized their cannon in exactly the same manner, yet the 
same underlying principle of calibers advancing in a geometrical 
series exists in all countries. This theory has withstood the most 
severe test that can be given any industrial product war. 

The design engineer is wasteful in production effort. He calcu- 
lates carefully the strength of parts of his design, and dimensions 

" Maj. Ord. Dept., U. S. A., Washington, D. C. Mem. A.S.M.E. 



DISCUSSION 53 

it to withstand exactly the stresses introduced. And to cover the 
inexactness of his formulas of computation, he introduces a factor 
of safety. There is no good reason why this factor cannot be mod- 
ified slightly to permit of the use of materials made according to 
definite lines of preferred numbers. One of our most careful engi- 
neers, when asked for his opinion on this paper, said of course it 
was good for other design work but could not be applied in such a 
careful designing as is required in ordnance construction. The 
recuperator, for example, requires special sizes for the piston. 
Upon its area depend the cylinder diameter and the thickness of 
walls of the cylinder. High pressures and comparatively low fac- 
tors of safety are used. But when this problem was carefully 
analyzed it was found that a variation of ^ in. in 6 in. made no 
practical difference in the design. 

In some recent standardization work the necessity for standard 
I 3 / 8 - and 2 3 /4-in. bolts was discussed. The percentage variation 
in area between 5 / 8 - and 3 A-in. bolts is the same as between V/ r 
and iVz-in. bolts or between 2 l / r and 3-in. bolts. If n /i 6 -in. are 
not needed, why are ! 3 /s and 2 3 / 4 ? The designer says that he must 
have V4-in. steps in large sizes and the producer says that he 
manufactures large quantities of ! 3 /8-in. bolts. The area of the 
IVr-in. bolt is 44 per cent greater than that of the lV 4 -in. bolt. 
It is doubtful if many designers could justify the use of the l 3 /Vin. 
size or of the l /rm. step in sizes above 2 l / 2 in. 

CARL J. OXFORD. 18 At present we have many articles which do 
not show a uniform rate of variation in size and for no apparent 
reason. 

As a practical example of a uniform system of standardization 
applied to one particular product, I might here call attention to 
a paper on the Standardization of Small Tools recently read at a 
regional meeting of the Society. Without being acquainted at 
that time with the system of preferred numbers, I proposed as part 
of a standardization program the elimination of many sizes of twist 
drills now in more or less common use. Without regard to lengths 
or other differences of design, it was found that from l / z in. down 
there are 137 sizes or diameters of drills regarded as standard 
articles. Some of these are practically identical in size but are desig- 
nated by different symbols, while others show a non-uniform rate 
of variation. A tentative standard was then proposed. This 
standard was based on a system of groups of sizes, each group 
varying in a straight geometrical progression from the ones pre- 
ceding and following, while the variation of sizes within a group 
was uniform. By this process 64 sizes were eliminated and 73 
sizes were retained. 

It is interesting to note the effect of applying the preferred- 
number system to this same range of sizes, and Table 7 has accord- 



18 Natl. Twist Drill & Tool Co., Detroit, Mich. Assoc-Mem. A.S.M.E. 



54 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

ingly been worked out using the ratio \/10 or 1.059. The nearest 
corresponding commercial size used at present has been inserted in 
each case, and it is noted that with exception of two sizes the ex- 
isting diameters are very close to the theoretical diameters. This 
would result in the retention of only 64 sizes, which still would be 
ample for all practical purposes, provided the designers and me- 
chanics could be educated to their use. 

The proposal of basing the preferred-number system on some root 
of 10 is a natural one when it is considered that it originated in 
the European countries using the metric system. It does not fol- 
low, however, that this system of ratios is best suited for the English 
system of measures. To agree with the latter system it might be 
better to base the ratios on some root of 12 as \/12, 2 \/12, etc. 
\/12 would give 12 proportionate sizes between one foot and one 
inch, and retain these basic units as an integral part of the system. 

In closing, it might be said that any system is better than no 
system at all, which is precisely the condition now existing in vari- 
ations of sizes of many manufactured articles. 

TABLE 7 STRAIGHT-SHANK DRILL SIZES ARRANGED BY PRE- 
FERRED NUMBERS 



Actual 


size 
now in 


Actual 


now in 


Actual 


now in 


Actual 


size 

now in 


size 


use 


size 


use 


size 


use 


size 


use 


0.502 


/i 


0.200 


8 


0.080 


46 


0.032 


67 


0.474 


none 


0.189 


12 


0.075 


48 


0.030 


69 


0.448 


29/ M 


0.179 


15 


0.071 


50 


0.028 


70 


0.423 


"/M 


0.169 


18 


0.067 


51 


0.026 


71 


0.399 


X 


0.159 


21 


0.063 


Vl6 


0.024 


73 


0.377 


Y 


0.150 


25 


0.060 


53 


0.023 


6 mm. 


0.356 


T 


0.142 


/64 


0.057 


none 


0.022 


74 


0.336 


R 


0.134 


29 


0.054 


54 


0.021 


75 


0.317 





0.127 


30 


0.051 


55 


0.020 


76 


0.300 


N 


0.119 


31 


0.048 


3/ M 


0.019 


5 mm. 


0.283 


v 


0.113 


33 


0.045 


56 


0.018 


77 


0.267 


1J /M 


0.107 


36 


0.042 


58 


0.017 


none 


0.252 


>/4 


0.101 


38 


0.040 


60 


0.016 


78 


0.238 


B 


0.095 


41 


0.038 


62 


0.015 


v 


0.225 


1 


0.090 


43 


0.036 


64 


0.014 


79- 


0.212 


3 


0.085 


44 


0.034 


65 


0.013 


80 



REGINALD TRAUTSCHOLD. IS In the German system of preferred 
numbers, in order to make the various terms constituting a series 
convenient to use, the expedient is resorted to of rounding them 
off. That is, the individual terms are simply approximations, 
the nearest convenient decimal fractions being selected. This 
seems to be an inherent weakness, necessitating an arbitrary stand- 
ard for "rounding" the terms. 

A series of numbers in geometrical regressions is as effective as 
one in geometrical progression in reality such a series is infinitely 
more effective, as it is all-embracing and can be carried on in- 
definitely and the "head" of such a system should be 1, not 10. A 
geometrical regression with 1 as a head can be carried out to several 
terms without evolving an unfamiliar or inconvenient fractional 

19 69 Edgemont Road, Montclair, N. J. 



DISCUSSION 55 

term. For example, taking 1 in. as the head, the regressive series 
will be V 2 , Y 4 , Vs, Vie, 1 /32, VM, Vm in., etc., all familiar fractions 
in English measurements. If 1 ft. or 12 in. is taken as the head, 
the regressive series is 6, 3, I 1 /?, 3 A in., etc. These are all "pre- 
ferred numbers" in the sense of the suggested German system, and 
it will be noted that the inch and foot series dovetail into one another. 
There is a common ratio between consecutive numbers, or terms, and 
in addition every second term in the series has a definite relation, 
also every third, every fourth, etc., so that the series includes 
"preferred numbers" for roots and powers as well. 

Other systems of English weights and measures show in general 
a similar regard for preferred-number units, so it would appear that 
the English systems are basically preferred-number arrangements 
of extreme flexibility. It may be true that our use of the English 
measures is not as effective as it might be. We have confused their 
economic worth by the introduction of arithmetical simplification 
in calculations, but this does not alter the soundness of their geo- 
metrical derivation. 

In the metric systems of measures arithmetical progression only 
is employed, and this is their chief weakness. The interest in pre- 
ferred numbers evidenced by the Germans and other nations em- 
ploying a metric system of measures confirms this statement, for 
it is an effort to introduce a system of accepted measures which 
will conform to the requirements for geometrical progression that 
has brought about the preferred-number movement. We already 
have the most flexible system of preferred numbers. Why adopt 
now a system with inherent weaknesses unless we propose to 
supplant the English system of measures by a metric system? 

In our efforts toward standardization it would seem a pity to 
lean now toward a system which, recognizing its limitations, is 
endeavoring to adopt certain of the advantages inherent to the Eng- 
lish system. 

It is quite beyond me to see how the adoption of a system of 
preferred numbers derived in a manner similar to that adopted by 
the Germans could more effectively, or even as effectively, bring 
these improvements about than a suitable selection of preferred 
numbers from the flexible, comprehensive, and familiar geometrical 
series upon which the English system of measures is based. A. 
standard selection of preferred numbers of English origin is to be 
strongly advocated, but this is a relatively simple matter compared 
to the complication of our present measures by the further adoption 
of units of metric derivation which would take years to introduce 
and entail a cost of billions. 

OSCAR B. BjoRGE. 20 An investigation has been conducted to 
see what may constitute an ideal series of hoisting engines from the 
smallest cylinders, 3 in. by 4 in., up to the largest cylinders, 14 in. 



20 Gen. Mgr., Willamette-Clyde Co., Portland, Ore. Mem. A.S.M.E. 



36 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

by 16 in., choosing the intermediate sizes by the system of pre- 
ferred numbers. The principal factors of hoisting engines are the 
cylinders, crankshaft pinions, drum gears, and drum barrels, and 
when these are determined the rope pull of the engine is fixed. 
After choosing a series of cylinder sizes I have assumed a size of 
pinion, gear, and drum for the smallest and the largest, and chosen 
the intermediate sizes by means of preferred numbers. As a re- 
sult of this a line of cylinders has been picked without fractional 
bores or strokes, the ratio of bore to stroke being a uniformly chang- 
ing one. The resulting hoists exhibit a fine progressing uniformity, 
and, strange as it may seem, the resulting load capacities are more 
suited to the requirements of hoisting engines, determined by the 
weights and capacities of buckets, elevator cages, and other weights 
to be hoisted, than in the existing lines of hoists. There is prob- 
ably no reason for this last coincidence, and yet might it not be 
true that if a series be correctly laid out, thereby obtaining uni- 
formity, that the load capacities would be more suited to the vary- 
ing loads for which hoists are used? 

ARTHUR BESSEY SMITH. 21 The automatic equipment of a 
telephone central office which we make is practically always cus- 
tom-made as to size and arrangement. It seems not to be possible 
to build central offices in graded sizes, because each customer trims 
his requirements to the least possible size, so as to save initial ex- 
pense. However, each office is made with a definite ultimate 
capacity in view, and it is possible that this might be worked out 
on a definite series and the customer induced to accept an ultimate 
which is in the series. 

The private automatic exchange has not yet been standardized, 
but may possibly be eventually. That such apparatus can be put 
on a geometrical series seems doubtful, because of the decimal 
nature of the switches of which it is made and the few sizes involved. 

We doubt the applicability of preferred numbers directly to 
interior telephone cables, for their sizes are rather closely related to 
the number of terminals in racks and boards. 

Storage batteries, charging machines, and wires and cables used 
to carry power current are all bought from other manufacturers. 
We must accept their sizes and select those which most closely 
fit our ultimate demand. Preferred numbers may be of real as- 
sistance here. 

HARRY M. RoESER. 22 Rolling mills have been producing stand- 
ard structural shapes for a number of years, and as there has been 
no prominent movement for restandardizing them it appears fair 
to assume that the shapes now rolled are those that human experi- 
ence has hit upon as being most efficient, particularly in regard to 



21 Chief Research Engineer, Automatic Electric Co., Chicago, HI. 
Assoc. Engr. Physicist, Bur. of Standards, Washington, D. C. Assoc- 
Mem A.S.M.E. 



DISCUSSION 57 

the economy of metal used to meet trade requirements for strength 
and minimum costs. This being the case, if the preferred-number 
system is to be used for dimensioning these shapes, in order to 
justify itself it must yield at least as great efficiency in the distri- 
bution of metal as the present system. 

The writer's opportunities confined the investigation only to 
structural channels which are now rolled in ten depths, viz., 3, 4, 
5, 6, 7, 8, 9, 10, 12, and 15 in. Flange widths and web thicknesses 
are graded in a linear relation with respect to the depth approx- 
imately as follows: 

Flange width = 0.90 + (0.17 X depth) 
Web thickness = 0.14 + (0.01 X depth) 

The above dimensions are for what are called "Minimum Stand- 
ard" Channels. It is the trade practice to roll for each depth a 
number of different sizes of web thickness and flange widths. 

After some preliminary calculation it was concluded that to 
reproduce the present series of channel depths by consecutive 
terms a series of the form 10 1 /* was impracticable and consequently 
the terms of the 10'/ 80 series that best fitted the depths were selected. 
These terms are the 38th, 48th, 56th, 62nd, 68th, 72nd, 76th, 80th, 
86th, and 94th. This practice may be considered not exactly 
consistent with the preferred-number system, which apparently 
demands that consecutive sizes be graded according to consecutive 
terms of a geometrical series. However, it is contended that the 
sizes are graded according to the terms of the 80-series. It merely 
happens that the sizes corresponding to some of the terms of the 80- 
series have no practical demand and are not manufactured. 

The procedure of determining the series to fit the flange widths 
was as follows: 

The series was assumed to be of the form 10 l ' /x , where z is a 
number to be determined and the value of i which gives a certain 
flange width must be the same value of i that gives the correspond- 
ing web depth in the 80-series. That is to say, if the 38th term of 
the 80-series gives a certain web depth, then the 38th term of the 
z-series must give the corresponding flange width. On this basis 
the value of x was computed by the method of least squares to 
fit most closely the existing web thicknesses. The value computed 
for x was 191. The value used in this investigation was 192. 

The method was repeated to determine the web thicknesses. 
The value of x for this series was 180. 

Properties of the preferred-number sections compared to similar 
sections in current use are set forth in Table 8 and show that 
there is an economical advantage in favor of the current system of 
shapes. 

ALBERT W. WHITNEY." While the American Engineering Stand- 

23 National Bureau of Casualty and Surety Underwriters, 120 W. 42nd 
St., New York, N. Y. 



58 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

aids Committee does not itself deal with the technical details of 
standardization work, it has been very glad to cooperate with The 
American Society of Mechanical Engineers in arranging for a dis- 
cussion of this important paper from the point of view of a number 
of industries. 

That there can be an optimum selection among the multiplicity 
of possible sizes of manufactured products that shall be upon such 
a fundamental basis as to be applicable over a wide range of products 
is one of the most important and hopeful facts in the field of stand- 
ardization. Such a judgment with regard to the possibility and im- 
portance of such a system of preferred numbers is not based solely 
upon a priori reasoning, for in Europe, where the principal develop- 
ments in this field have taken place, actual results are showing both 
the importance and the feasibility of making use of such a selection : 
In France and in Germany, for instance, the development and appli- 
cation of preferred numbers is regarded as one of the most important 
achievements in the field of standardization. 

The choice of a number series is a matter that calls for something 
more than empirical treatment. The fact that for a wide variety 
of purposes the best series of sizes is geometrical is most interesting 
and significant. One cannot help feeling that this is only another 
manifestation of those facts of human nature which are expressed 
by the Fechner- Weber law. This law, it will be remembered, states 
that the increment of stimulus that produces an increment of sen- 
sation is proportional to the amount of the existing stimulus; for 
instance, if when I am holding a pound weight I can just notice 
the addition of an ounce, then when I hold a two-pound weight I 
shall be just able to notice the addition of two ounces. 

The fact is that in the type of preferred-number series that is 
under consideration we have apparently got hold of something that 
is fundamental: it has far more sanction than mere empiricism. 

It has, for instance, an earmark that is found in every fundamental 
solution, namely, that it clears up a larger field than we had hoped 
initially to affect. As illustrating this I will instance the facts: 
first, that the number series that obtains where increased refinement 
is called for can easily be made to include the numbers in all lower 
series; second, that if linear dimensions are expressible by a geo- 
metrical series, then areas and volumes are also expressible by num- 
ber series which are contained in the original series; and, third, 
by the apparent adaptability, due to fundamental considerations, 
of such number series to the simplification of machine design. 

One other important fact should be noted. The fundamental 
character of this method would adapt it for use as a basis for inter- 
national standardization. In fact, it has been suggested by two 
foreign national standardizing bodies that no time should be lost 
in getting the countries of the world together, not only in agree- 
ment upon a particular number series to be used, but upon a stand- 
ardization of roundings. If such a common number series were 



DISCUSSION 



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60 SIZE STANDARDIZATION BY PREFERRED NUMBERS 

individual who has discussed the paper, but shall content myself 
with replying to the discussion in a very general way. First, 
standardization is arbitrary. That is the basis on which it starts. 
The question is this: If we give due consideration to the proper 
claims of all interested parties, is this arbitrary standardization 
worth while or not worth while? So far as we know, judging from 
human experience, I believe the majority of people think that it 
is worth while. 

If it is worth while, and it must be arbitrary, the question is 
really, What does such arbitrary standardization lead to in the re- 
lations between supplier and user? 

I will grant that the commercial tendency is to supply all the 
demands of the users and, as one would expect, a consequent large 
demand. 

I happen to represent two organizations which last year under- 
took to standardize the steam turbine. We started by trying to 
standardize sizes of the steam turbine. We had the makers with 
us, and the makers were as usual between the devil and the deep 
sea. The users demanded certain very closely spaced sizes, and 
the manufacturers naturally wanted to meet the users' demands. 
On the other hand, the users were complaining about the cost of 
the unit. 

It is perfectly obvious that if a man makes six sizes over a given 
range instead of twelve, the units will be cheaper in the case of such 
things as steam turbines; and we finally agreed upon certain stand- 
ards, and sooner or later, if I am not mistaken, those sizes will be 
the standards of steam turbines in this country. 

It does not mean that you cannot buy other sizes, but it means 
that when you want a steam turbine you will buy the standard or 
you will pay the burden that you put on the industry. 

I think that this is the keynote in the adjustment between user 
and supplier. Give the user what he wants if he is willing to pay 
for it, and you will soon find that in most cases he does not want it- 
There are some figures in the paper, notably Fig. 8, which show 
the results of an attempt to meet the needs, or supposed needs, of 
everybody. 

I do not know whether they are justifiable needs or not; I do not 
pretend to know, but I do know that the sizes there shown are so 
closely spaced that there is at least the inference that they are not 
all necessary. 

Considerable discussion has been given to the question of what 
series we should use. I think that is of very little importance right 
now. The question is, Do we want to use a series? Do we want to 
use preferred numbers? And if we do, should we arrange them in 
a geometrical series? If we do not use such a series, what one do 
we want to use for the preferred-number series? That is entirely 
& separate question from the matter of units. 

I am not prepared to say that the 80th root of 10 is the ruling one, 



DISCUSSION 61 

or the 12th root of 12 is the best. I do not know enough about it. 
But if you first decide that there is enough in preferred numbers 
to make an effort to secure their adoption, the rest of the problem 
will take care of itself. With a few men like Mr. Earth around, 
we will get the system that fits. 

I understood Mr. Speed to say that he was totally -opposed to 
every standardization. I do not know whether I correctly quote 
him or not, but if that does express his attitude, I think possibly it 
comes from the fact that in the industry in which he works every- 
thing would lean toward that sort of a conclusion. 

However, with this I am satisfied : It is not a question of whether 
some of us want standardization and some of us do not. Standard- 
ization is forced on us. If we are going to survive commercially 
we must standardize, and if we must standardize, let us get off 
and view the problem from the greatest possible distance. Con- 
sider it sanely and cold-bloodedly and decide we are going to stand- 
ardize this way or that way, and after we have done that, having 
decided which way we are going to standardize not whether we 
will or not, because we have got to then let us discuss the details 
of that standardization. 



'- 



